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arXiv:math/0407298v1 [math.AC] 16 Jul 2004
TETRAHE D RAL CURVES
J. MIGLIORE
∗
, U. NAGEL
+
Abstract. A tetrahedral curve is a space curve whose defining ideal is an intersection
of powers of monomial prime ideals of height two. It is suppor ted on a tetrahedral
configuration of lines. Schwartau described when certain such curves are ACM, namely
he restricted to curves supported on a certain four of the six lines. We consider the
general situation.
We first show that starting with an arbitrary tetrahedral curve, there is a particular
reduction that produces a smaller tetrahedral curve and preserves the even liaison class.
We call the curves that are minimal w ith respect to this reduction S-minimal curves.
Given a tetrahedral curve, we describe a simple alg orithm (involving only integers) that
computes the S-minimal curve of the corresponding even liaison cla ss; in the process
it determines if the original curve is arithmetically Cohen-Macaulay or not. We also
describe the minimal free resolution of an S-minimal curve, us ing the theory of cellular
resolutions. This resolution is always linear. This result allows us to class ify the arith-
metically Buchsbaum, non-ACM tetrahedral curves. More importantly, it allows us to
conclude that a n S-minimal curve is minimal in its even liaison class; that is, the whole
even liaison class can be built up from the S-minimal curve. Finally, we show that there
is a large set of S-minimal curves such that each curve corresponds to a smooth point of
a component of the Hilbert scheme and that this compo nent has the expected dimension.
Contents
1. Intro duction 2
2. Background 3
3. Reduction and S-minimality 5
4. The Minimal Free Resolution of an S-minimal Curve 9
5. Minimality in the Even Liaison Class, and Applications 14
6. Unobstructedness of some curves 19
7. Remarks and problems 25
Appendix: The algorithm to find S-minimal curves 27
References 30
∗
Part of the work for this paper was done while this author was sponsored by the National Security
Agency under Grant Number MDA904-03-1 -0071.
+
This a uthor gratefully acknowledges partial support by a Special Faculty Research Fellowship from the
University of Kentucky.
1
2 J. MIGLIORE, U. NAGEL
1. Introduction
In his Ph.D. thesis [20], which was never published, Phil Schwartau considered certain
monomial ideals and the question of whether or not they were Cohen-Macaulay. Specifi-
cally, he considered ideals of the form
(X
0
, X
1
)
a
∩ (X
1
, X
2
)
b
∩ (X
2
, X
3
)
c
∩ (X
3
, X
0
)
d
in the ring k[X
0
, X
1
, X
2
, X
3
] where k is a n algebraically closed field. These are unmixed
ideals defining curves in P
3
. They are supported on a complete intersection (taking
(a, b, c, d) = (1, 1, 1, 1) gives the complete intersection of X
0
X
2
and X
1
X
3
). Schwartau
gave a complete classification of the 4-tuples of integers (a, b, c, d) that define arithmeti-
cally Cohen-Macaulay curves (see Theorem 2.4 and Theorem 5.3).
This paper arose from our desire to make the natural extension of this result to include
the lines defined by (X
0
, X
2
) and (X
1
, X
3
). To simplify the notation we have changed the
var ia bles, so we are considering the r ing R = k[a, b, c, d] and ideals of the form
I = (a, b)
a
1
∩ (a, c)
a
2
∩ (a, d)
a
3
∩ (b, c)
a
4
∩ (b, d)
a
5
∩ (c, d)
a
6
.
Note that I is an unmixed monomial ideal. Since these six lines can be viewed as forming
the edges of a tetrahedron, we call such a curve a tetrahedral c urve. It is useful to consider
the empty set as the trivial curve defined by (0, 0, 0, 0, 0, 0).
More than simply considering the question of when such a curve is arithmetically Cohen-
Macaulay, we are interested in getting some idea of the even liaison class of such a curve.
For background on liaison, see the book [12]; fo r the most part we will assume the necessary
definitions and basic results on liaison.
Many papers in the literature give classification results of the following kind: they
consider a particular kind of curve, and ask when two such curves are linked, or ask
for a description of the even liaison class of such a curve (cf. for instance [5], [13], [14],
[18]). The work in this paper can be viewed, in part, as a new contribution to this kind
of question. However, the main interest comes from the naturalness of these monomial
ideals themselves, from the surprising effectiveness of our reduction procedure, from the
simplicity of the minimal free resolution obtained, and from the interplay of strikingly
different techniques and tools used to obtain these results.
The key idea that got this work started was a realization that there is a simple reduction
possible for tetrahedral curves (Proposition 3.1 ) . Using the machinery of basic double
links, this reduction accomplishes two amazing things: the new curve is in the same even
liaison class a s the original curve, and the new curve is again a tetrahedral curve, but
smaller!
Using this reduction, one of two things happens: either the process o f reduction contin-
ues until the curve vanishes, i.e. we get the trivial curve, or the process stops at a curve
that cannot be further reduced in this way. The curves that ultimately reduce to the triv-
ial curve are clearly arithmetically Cohen-Macaulay, thanks to liaison theory. Curves that
cannot be further reduced in this way are very important, and we call them S-minimal
curves. We give a numerical criterion fo r S-minimality (Corollary 3.5 and Lemma 3.8).
This leads to a simple a lg orithm (easily done by hand) for computing an S-minimal curve
starting with any tetrahedral curve. A MAPLE implementation of it is documented in
the appendix.
TETRAHEDRAL CURVES 3
The first main result of this paper is Theorem 4.2 , where the minimal free resolution of
an S-minimal tetrahedral curve is computed. We show, in particular, that this resolution
is linear. This proo f starts by finding the minimal generators of the ideal, and then it
translates these generators into a cell complex and uses the theory of cellular resolutions
as developed in [2] to find the rest of the resolution.
An immediate consequence of this result is that an S-minimal curve is arithmetically
Cohen-Macaulay if and only if it is trivial. This allows one to easily determine, for
any 6-tuple of integers (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
), whether it corresponds to an arithmetically
Cohen-Macaulay curve or not. Another easy consequence is that a tetrahedral curve
is arithmetically Buchsbaum if and only if its Hartshorne-Rao module has diameter 1.
(It was already known by Schwartau that certain arithmetically Buchsbaum tetrahedral
curves existed.)
A much deeper consequence of Theorem 4.2 is the second main result of the paper
(Theorem 5.1), which says that a tetrahedral curve C is S-minimal if and only if it is
minimal in its even liaison class.
We give two applications of this result. The first, Theorem 5.3, is a new proof of
Schwartau’s theorem classifying the arithmetically Cohen-Macaulay curves supported on
the complete intersection by giving the precise set of 4-tuples. The second application,
Corollary 5.4, is a classification of the 6-tuples that define minimal tetrahedral curves
that are arithmetically Buchsbaum. Both of these applications primarily use the earlier
results described above, but need the minimality in the even liaison class to complete the
proof.
Our third main line of investigation concerns the question of unobstructedness and the
search for nice components of the Hilbert scheme. (A component is said to be “nice” if
it has the expected dimension 4 · deg C.) A great deal of work has been done concerning
nice components of the Hilbert scheme – we refer the reader to [7] and to [9], both
for important results along these lines, and a lso for numerous references to other work.
Using the method of Dolcetti in [7], we first prove that any curve in P
3
with linear
resolution and Hartshorne-Rao module of diameter ≤ 2 is unobstructed, and its Hilbert
scheme has the expected dimension 4 · deg C (Proposition 6.1). We then characterize t he
6-tuples corresponding to minimal tetrahedral curves with Hartshorne-Rao modules of
diameter ≤ 2 (Corollary 5.4 and Lemma 6.2), which then are unobstructed. Turning to
a different kind of minimal tetrahedral curve, we show that a tetrahedral curve defined
by (a
1
, 0, 0, 0, 0, a
6
) is also unobstructed. Since we have shown the unobstructedness of a
large class of minimal tetrahedral curves, and have shown experimentally on the computer
that others are also unobstructed, we end with the question of whether in fact a ll minimal
tetrahedral curves are unobstructed.
We end the paper with a number of questions that arise naturally from our work.
2. Background
Let R = k[a, b, c, d], where k is a field. We abbreviate by ACM the term “a r ithmeti-
cally Cohen-Macaulay.” Recall that a projective subscheme V is ACM if and only if the
deficiency modules all vanish:
H
i
(P
n
, I
V
(t)) = 0 for all t ∈ Z and a ll 1 ≤ i ≤ dim V .
4 J. MIGLIORE, U. NAGEL
We recall that the notion of basic double linkag e was intr oduced by Lazarsfeld and Rao
[10] as a way of adjoining a plane curve, or more generally a complete intersection, to a
given curve C in such a way as to preserve the even liaison class of C. More precisely,
Definition 2.1. Let I ⊂ R be the saturated ideal of an unmixed curve C in P
3
. Let
F ∈ I and G ∈ R be homogeneous polynomials such that (F, G) is a regular sequence.
Then the ideal G · I + (F ) is the saturated ideal of a curve Y which is linked to C in
two steps (i.e. is bil i nked to C). Y is said to be a basic double link of C. As sets, Y
is the union of C and the complete intersection defined by (F, G). The degree of Y is
deg C + (deg F )(deg G).
The notion of basic do uble linkage has generalizations in several different directions
(higher dimension, higher projective spaces, Go r enstein liaison). We refer to [12] for the
details. If one has a way of recognizing an ideal as being of the above for m, then one can
replace that ideal by the simpler ideal I, knowing that the liaison class is preserved. For
instance, if the original ideal is Cohen-Macaulay then so is the new, simpler ideal. This
is the approach we take below.
Recall that for an unmixed ideal I, the n-th symbolic power of I, denoted I
(n)
, is the
saturation of the t op dimensional part of the ideal I
n
. Recall also that if I is a complete
intersection then I
(n)
= I
n
.
Definition 2.2. Let T be any union of six lines ℓ
1
, . . . , ℓ
6
forming the edges of a “tetra-
hedron.” A tetrahedral curve is the non- r educed scheme C supported on T and defined
by the saturated ideal I
C
= I
a
1
ℓ
1
∩ · · · ∩ I
a
6
ℓ
6
, where a
i
≥ 0 for all i. The number a
i
is called
the weight of the line l
i
.
Remark 2.3. If C is a tetrahedral curve then we can perform a change of variables and
obtain a monomial ideal which has a primary decomposition of the form
I = (a, b)
a
1
∩ (a, c)
a
2
∩ (a, d)
a
3
∩ (b, c)
a
4
∩ (b, d)
a
5
∩ (c, d)
a
6
.
For the remainder of this paper we will assume that a tetrahedral curve is a monomial
ideal of this form.
In his thesis [20], Schwartau considered the curves with ideals of the form
(X
0
, X
1
)
a
∩ (X
1
, X
2
)
b
∩ (X
2
, X
3
)
c
∩ (X
3
, X
0
)
d
(in his notation). This is equivalent to taking a
2
= a
5
= 0 in Definition 2.2. Schwartau
was primarily interested in the question of when these curves are arithmetically Cohen-
Macaulay. His main result on this problem is the following, which we now translate to
our language.
Theorem 2.4 ([20]). The ideal
(a, b)
a
1
∩ (a, c)
0
∩ (a, d)
a
3
∩ (b, c)
a
4
∩ (b, d)
0
∩ (c, d)
a
6
defines an arithmetically Cohen-Macaulay curve in P
3
if and only if
Case 1
. a
1
, a
3
, a
4
, a
6
> 0: a
1
+ a
6
= a
3
+ a
4
+ ǫ, f or ǫ = −1, 0, 1.
Case 2
. a
1
, a
4
, a
6
> 0, a
3
= 0: a
1
+ a
6
≤ a
4
+ 1.
Case 3A
. a
1
, a
4
> 0, a
3
= a
6
= 0: always
TETRAHEDRAL CURVES 5
Case 3B. a
1
, a
6
> 0, a
3
= a
4
= 0: never
Case 4
. a
1
> 0, a
3
= a
4
= a
6
= 0: always
Remark 2.5. Clearly Schwartau intended some kind of reduction of cases, since for
instance the case a
2
> 0, a
1
= a
3
= a
4
= 0 is not included in his theorem. We have
given an “invariant” version in Theorem 5.3 below, which we think reflects Schwartau’s
intention. We also give a new proof .
3. Reduction and S-minimality
The key to our approach to this problem is the following reduction method.
Proposition 3.1. Let I = (a, b)
a
1
∩ ( a, c)
a
2
∩ ( a, d)
a
3
∩ ( b, c)
a
4
∩ ( b, d)
a
5
∩ ( c, d)
a
6
where
not all exponents a
i
are zero. Cons i der the following systems of inequalities:
(A) : a
1
+ a
2
≥ a
4
,
a
1
+ a
3
≥ a
5
,
a
2
+ a
3
≥ a
6
(B) : a
1
+ a
4
≥ a
2
,
a
1
+ a
5
≥ a
3
,
a
4
+ a
5
≥ a
6
(C) : a
2
+ a
4
≥ a
1
,
a
2
+ a
6
≥ a
3
,
a
4
+ a
6
≥ a
5
(D) : a
3
+ a
5
≥ a
1
,
a
3
+ a
6
≥ a
2
,
a
5
+ a
6
≥ a
4
.
For 1 ≤ i ≤ 6 let a
′
i
= max{0, a
i
− 1}. Then we have
(i) (A) ⇔ I is a bas i c d o uble li nk of
(a, b)
a
′
1
∩ (a, c)
a
′
2
∩ (a, d)
a
′
3
∩ (b, c)
a
4
∩ (b, d)
a
5
∩ (c, d)
a
6
using F = b
a
1
c
a
2
d
a
3
and G = a.
(ii) (B) ⇔ I is a basic double link of
(a, b)
a
′
1
∩ (a, c)
a
2
∩ (a, d)
a
3
∩ (b, c)
a
′
4
∩ (b, d)
a
′
5
∩ (c, d)
a
6
.
using F = a
a
1
c
a
4
d
a
5
and G = b.
(iii) (C) ⇔ I is a basic double link of
(a, b)
a
1
∩ (a, c)
a
′
2
∩ (a, d)
a
3
∩ (b, c)
a
′
4
∩ (b, d)
a
5
∩ (c, d)
a
′
6
.
using F = a
a
2
b
a
4
d
a
6
and G = c.
(iv) (D) ⇔ I is a basic double link of
(a, b)
a
1
∩ (a, c)
a
2
∩ (a, d)
a
′
3
∩ (b, c)
a
4
∩ (b, d)
a
′
5
∩ (c, d)
a
′
6
.
using F = a
a
3
b
a
5
c
a
6
and G = d.
Proo f . We will prove (i); of course the others are proved similarly. First note that
a · (a, b)
n−1
+ (b
n
) = (a, b)
n
for n ≥ 1. Now consider the monomial F = b
a
1
c
a
2
d
a
3
. Notice that
(3.1) F ∈ (a, b)
a
1
∩ (a, c)
a
2
∩ (a, d)
a
3
⊂ (a, b)
a
′
1
∩ (a, c)
a
′
2
∩ (a, d)
a
′
3
even when one or more of the a
i
= 0. The three inequalities a re equivalent to
(3.2) F ∈ (b, c)
a
4
∩ (b, d)
a
5
∩ (c, d)
a
6
.
6 J. MIGLIORE, U. NAGEL
Hence
F ∈ (a, b)
a
′
1
∩ (a, c)
a
′
2
∩ (a, d)
a
′
3
∩ (b, c)
a
4
∩ (b, d)
a
5
∩ (c, d)
a
6
.
Notice that (a, F ) is a regular sequence. Hence we can construct a basic double link of
the form
J = a ·
h
(a, b)
a
′
1
∩ (a, c)
a
′
2
∩ (a, d)
a
′
3
∩ (b, c)
a
4
∩ (b, d)
a
5
∩ (c, d)
a
6
i
+ (b
a
1
c
a
2
d
a
3
).
We have t o show that
J = I = (a, b)
a
1
∩ (a, c)
a
2
∩ (a, d)
a
3
∩ (b, c)
a
4
∩ (b, d)
a
5
∩ (c, d)
a
6
.
The ideal J is a saturated, unmixed ideal, by the theory of basic double linkage (cf. [12]),
so it is enough to show that J ⊂ I and that they define schemes of the same degree.
For the first one, the fact that
a ·
(a, b)
a
′
1
∩ (a, c)
a
′
2
∩ (a, d)
a
′
3
∩ (b, c)
a
4
∩ (b, d)
a
5
∩ (c, d)
a
6
⊆ (a, b)
a
1
∩ (a, c)
a
2
∩ (a, d)
a
3
∩ (b, c)
a
4
∩ (b, d)
a
5
∩ (c, d)
a
6
is clear, while the fact that b
a
1
c
a
2
d
a
3
∈ I comes from (3.1) and (3.2) above.
For the degree computation, recall that
n
2
+ n =
n+1
2
. Then, ag ain using the theory
of basic double linkage, the degree of J is
deg J =
a
1
2
+
a
2
2
+
a
3
2
+
a
4
+ 1
2
+
a
5
+ 1
2
+
a
6
+ 1
2
+ (a
1
+ a
2
+ a
3
)
=
a
1
+ 1
2
+
a
2
+ 1
2
+
a
3
+ 1
2
+
a
4
+ 1
2
+
a
5
+ 1
2
+
a
6
+ 1
2
= deg I
as desired.
The converse is immediate: the fact that I is a basic double link as stated implies the
inequalities of (A) by again using(3.1) and (3.2).
Notation 3.2. For the r est of this paper we will abbreviate the monomial ideal
(a, b)
a
1
∩ (a, c)
a
2
∩ (a, d)
a
3
∩ (b, c)
a
4
∩ (b, d)
a
5
∩ (c, d)
a
6
(or the corresponding curve) by the 6-tuple (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
).
Remark 3.3. Basic double linkag e is a special case of Schwartau’s “liaison addition.”
(See [20] for the or ig inal and [8] for a generalization.) Without entering into details, we
remark that many interesting tetrahedral curves arise as liaison additions. For instance,
let I be the ideal of the six lines, i.e. the tetrahedral curve (1, 1, 1 , 1, 1, 1). Let F be the
polynomial abcd giving the four faces of t he “tetrahedron” defined by I. Note that F is
double along each of the six lines. Let G be a generally chosen cubic in I. Then (F, G)
self-links I (as can be see geometrically, using Bezout’s theorem) and one can check t hat
in fact the d-th symbolic power of I, I
(d)
, can be expressed as
I
(d)
= G
d−1
· I + F · I
(d−2)
.
Hence we see directly the curves ( d, d, d, d, d, d) are ACM. Note however, that the d-th
power I
d
is not saturated, and has embedded points.
TETRAHEDRAL CURVES 7
Schwartau also o bta ins many Buchsbaum curves using liaison addition. We will return
to Buchsbaum curves below.
Definition 3.4. We say that a tetrahedral curve is S-minimal if there is no reduction of
the typ e described in Proposition 3.1.
Corollary 3.5. Consider a tetrahedral curve C = (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) where not all a
i
are 0. Assume without l oss of generality that a
6
= max{a
1
, . . . , a
6
}. Then C is S-minimal
if and only if
a
1
> max{a
3
+ a
5
, a
2
+ a
4
} and
a
6
> max{a
4
+ a
5
, a
2
+ a
3
}
Proo f . It is immediate to check that if the stated conditions are satisfied then each of (A),
(B), (C) and ( D) in Proposition 3.1 has at least one inequality that is not satisfied, hence
C cannot be reduced via Proposition 3.1.
Conversely, suppose that C is S-minimal. The second and third inequalities of (C) and
(D) in Proposition 3 .1 are forced to be true by the assumption that a
6
= max{a
1
, . . . , a
6
},
so S-minimal implies that a
1
> max {a
3
+ a
5
, a
2
+ a
4
}. In particular, a
1
> a
2
, a
1
> a
3
,
a
1
> a
4
and a
1
> a
5
, so the first two inequalities of (A) and of (B) must be true. Hence
again the assumption that C is S-minimal forces a
6
> max{a
4
+ a
5
, a
2
+ a
3
}.
Example 3.6. Even with the assumption that a
6
= max{a
1
, . . . , a
6
}, the first inequal-
ity of Corollary 3.5 alone does not imply S-minimality, as shown by the example C =
(4, 2, 2, 1, 1, 4), which can be reduced using (A) to the curve C
′
= (3, 1, 1, 1, 1, 4). This
new curve C
′
is S-minimal. It can be checked that C
′
(and hence C) is not arithmetically
Cohen-Macaulay (see Corollary 4.3 as well).
Our next goal is to provide an algorithm that produces an S-minimal curve to a given
tetrahedral curve C. To this end we will extend the definition of weights. We will use the
tetrahedron T = T (C) whose edges are part of the (po tentially) supporting lines of the
curve C.
Definition 3.7. The weight of a facet o f T = T (C) is the sum of the weights of the edges
forming its boundary. Similarly, the weight of a pair of skew lines being a subset of the
six given lines is the sum of the weights of the lines.
Using these concepts we can make Corollary 3.5 more precise. Note that each of the
reductions (A) - (D) in Proposition 3.1 reduces the weights of the edges of a facet of
the tetrahedron. Thus, we will say that we reduce a facet if we a pply the corresponding
reduction.
The next result says in particular that a tetrahedral curve is not minimal if and only if
we can reduce a facet of maximal weight.
Lemma 3.8. Let C be a tetrahedral curve C = (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) where we ass ume
without loss of generality that a
6
= max{a
1
, . . . , a
6
} > 0. Let w be the maximal weight of
a facet of the tetrahedron T = T (C). Then the following conditions are equivalent:
(a) C is not S-minimal.
(b) a
1
+ a
6
≤ w.
(c) One can red uce any of the facets of T havin g m aximal weight w.
8 J. MIGLIORE, U. NAGEL
Proo f . We begin by showing the equivalence of ( a) and (b). According to Corollary 3.5
we know that C is not S-minimal if and only if
a
1
≤ max{a
3
+ a
5
, a
2
+ a
4
} or
a
6
≤ max{a
4
+ a
5
, a
2
+ a
3
}.
But this is equivalent to the condition
a
1
+ a
6
≤ max{a
1
+ a
4
+ a
5
, a
1
+ a
2
+ a
3
, a
3
+ a
5
+ a
6
, a
2
+ a
4
+ a
6
} = w
as claimed.
Since (c) tr ivially implies (a), it remains to show that (c) is a consequence of (b). To
this end we distinguish four cases.
Case 1: Let w = a
3
+ a
5
+ a
6
. Then (b) provides a
3
+ a
5
≥ a
1
showing that we can
apply reduction (D).
Case 2: Let w = a
2
+ a
4
+ a
6
. Then we conclude as above that we can use reduction
(C).
Case 3: Let w = a
1
+ a
2
+ a
3
. Then (b) implies a
2
+ a
3
≥ a
6
. Moreover, we have by
assumption w ≥ a
2
+ a
4
+ a
6
. It implies
a
1
+ a
3
≥ a
4
+ a
6
≥ a
6
≥ a
5
.
Similarly w ≥ a
3
+ a
5
+ a
6
provides a
1
+ a
2
≥ a
4
. Thus, we have shown that we can apply
reduction (A).
Case 4: Let w = a
1
+ a
4
+ a
5
. Then we see as in Case 3 that we can use reduction
(B).
The last result leads to an algorithm for producing S-minimal curves that works with
weights only.
Algorithm 3.9 (for computing S-minimal curves).
Input: (a
1
, . . . , a
6
) ∈ Z
6
where all a
i
≥ 0, the weight vector of a tetrahedral curve C.
Output: (a
1
, . . . , a
6
), the weight vector of an S-minimal curve obtained by reducing C.
1. compute i such that a
i
= max{a
1
, . . . , a
6
}
2. if a
i
= 0 then return (0, . . . , 0)
3. determine a facet F of maximal weight w
4. if a
i
+ a
7−i
> w then return (a
1
, . . . , a
6
).
5. apply the r eduction corresponding to the f acet F and go to Step 1
Remark 3.10. (i) Strictly speaking the scheme above is not an algor ithm since it leaves
choices in Steps 1 and 3 in case there is more than one line or facet of maximal weight.
But this can be fixed easily, e.g., by using the lexicographic order. See the appendix.
TETRAHEDRAL CURVES 9
(ii) Correctness of Algorithm 3.9 follows by Lemma 3.8. Note that the lines with index
i and 7 − i do not intersect.
4. The Minimal Free Resolution of an S-minimal Curve
Notice that the 6-tuple (0, . . . , 0) corresponds to the ring R. It turns out that it is
useful to consider R formally as a curve as we did in Definition 2.2. We g ive it a name.
Definition 4.1. The trivial tetrahedral curve is defined by (0, . . . , 0).
The following is one of the two main results of this paper.
Theorem 4.2. Every non-trivial S- minimal tetrahedra l curve has a linear minimal free
resolution.
More p recisely, if the curve C is defined by (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) and a
6
= max{a
i
} > 0
then its min i mal free resolution has the form
0 → R
β
3
(−a
1
− a
6
− 2) → R
β
2
(−a
1
− a
6
− 1) → R
β
1
(−a
1
− a
6
) → I
C
→ 0
where
β
1
= (a
1
+ 1)(a
6
+ 1) −
5
X
i=2
a
i
(a
i
+ 1)
2
β
2
= 2a
1
a
6
+ a
1
+ a
6
−
5
X
i=2
a
i
(a
i
+ 1)
β
3
= a
1
a
6
−
5
X
i=2
a
i
(a
i
+ 1)
2
.
Proo f . Let C be an S-minimal tetrahedral curve, and assume that C is the curve defined
by the tuple (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
). Without loss of generality assume that a
6
is the largest
entry. Note that (a, b)
a
1
and (c, d)
a
6
are disjoint ACM curves, so their intersection is given
by their product (cf. [21], Corollaire, p. 1 43 or [15], Theorem 1.2). In particular, the curve
(a
1
, 0, 0, 0, 0, a
6
) is given by the entries of the following (a
1
+ 1) × (a
6
+ 1) matrix:
(4.1)
a
a
1
c
a
6
a
a
1
c
a
6
−1
d . . . a
a
1
cd
a
6
−1
a
a
1
d
a
6
a
a
1
−1
bc
a
6
a
a
1
−1
bc
a
6
−1
d . . . a
a
1
−1
bcd
a
6
−1
a
a
1
−1
bd
a
6
.
.
.
.
.
.
.
.
.
.
.
.
ab
a
1
−1
c
a
6
ab
a
1
−1
c
a
6
−1
d . . . ab
a
1
−1
cd
a
6
−1
ab
a
1
−1
d
a
6
b
a
1
c
a
6
b
a
1
c
a
6
−1
d . . . b
a
1
cd
a
6
−1
b
a
1
d
a
6
By Corollary 3.5, we have t he inequalities
(4.2)
a
1
> a
3
+ a
5
a
1
> a
2
+ a
4
a
6
> a
4
+ a
5
a
6
> a
2
+ a
3
In particular, a
1
> 0.
10 J. MIGLIORE, U. NAGEL
Now we want to describe the ideal (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
). Note that as ideals,
(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) ⊂ (a
1
, 0, 0, 0, 0, a
6
).
Consider the monomials obtained by deleting the following from the matrix (4.1): a
2
diagonals from the “Southeast corner,” a
3
diagonals from the “Southwest corner,” a
4
diagonals from t he “No r theast corner,” and a
5
diagonals from t he “No r thwest corner.”
We obtain t he following shape:
(4.3)
a
1
@
@
@
@
@
@
a
3
a
3
a
5
a
5
a
4
a
2
a
2
a
4
|
{z }
a
6
Note that the a
i
in the diagram measure the length, and not the number of vertices. Note
also that the minimality, and in particular the inequalities (4.2), guarantee that these
“cuts” do not overlap. In particular, any monomial that is removed falls unambiguously
into exactly one of the removed regions.
Now let us define I(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) to be the ideal generated by the remaining
monomials, a f t er removing the corners as described above. We first make the fo llowing
claim:
Claim: I(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) = (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
), that is that I(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) is
precisely the ideal of the corresponding tetrahedral curve.
It is clear that the monomial generators that we have removed from (a
1
, 0, 0, 0, 0, a
6
)
are precisely those generators that are not in (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
). This proves ⊆.
We now prove the reverse inclusion. Since (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) is a monomial ideal and
(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) ⊂ (a
1
, 0, 0, 0, 0, a
6
), we see that the two ideals in the statement of the
claim agree in degrees ≤ a
1
+ a
6
. The danger is that (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) could have a
monomial minimal generator of larger degree. We now show that this does not occur.
Suppose that M were such a generator, of degree > a
1
+a
6
. M cannot be a multiple of a
minimal g enerator of I(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
), by the first inclusion. But on the other hand,
M is contained in (a
1
, 0, 0, 0, 0, a
6
). Therefore M is a multiple of one of the monomials
that we have removed, say N.
We will consider the case where N is in the r emoved “Northwest corner;” the other cases
are identical. We have removed a
5
diagonals from this corner, where a
5
is the exponent
of the component (b, d). Clearly if we multiply N by either a or c, we do not produce a
monomial that is in (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
). (The problem is in the component (b, d).)
TETRAHEDRAL CURVES 11
More generally, suppose that N lies o n the k-th diagonal from the “Northwest corner”
(k ≤ a
5
). Then N has the form a
ℓ
1
b
ℓ
2
c
ℓ
3
d
ℓ
4
where
(4.4)
ℓ
1
+ ℓ
2
= a
1
,
ℓ
3
+ ℓ
4
= a
6
,
ℓ
2
+ ℓ
4
= k − 1 (< a
5
)
Suppose that we multiply N by a monomial a
k
1
b
k
2
c
k
3
d
k
4
. The result is, of course,
a
k
1
+ℓ
1
b
k
2
+ℓ
2
c
k
3
+ℓ
3
d
k
4
+ℓ
4
. This will be in (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) if and only if
(4.5)
k
1
+ ℓ
1
+ k
2
+ ℓ
2
≥ a
1
k
1
+ ℓ
1
+ k
3
+ ℓ
3
≥ a
2
k
1
+ ℓ
1
+ k
4
+ ℓ
4
≥ a
3
k
2
+ ℓ
2
+ k
3
+ ℓ
3
≥ a
4
k
2
+ ℓ
2
+ k
4
+ ℓ
4
≥ a
5
k
3
+ ℓ
3
+ k
4
+ ℓ
4
≥ a
6
.
We make the subclaim that (4.5) holds if and only if k
2
+ k
4
≥ a
5
− (k − 1). If (4.5)
holds then from (4.5) and (4.4) respectively we have
k
2
+ k
4
≥ a
5
− ℓ
2
− ℓ
4
= a
5
− (k − 1)
as desired. Conversely, assume that k
2
+ k
4
≥ a
5
− (k − 1). The first and last inequalities
of (4.5) are immediate from (4.4), without needing our hypothesis. Similarly, none of the
other inequalities apart from the fifth one need our hypothesis. Consider for instance the
second inequality. We have
k
1
+ ℓ
1
+ k
3
+ ℓ
3
= k
1
+ k
3
+ a
1
+ a
6
− (k − 1) (from (4.4))
> k
1
+ k
3
+ a
2
+ a
4
+ a
4
+ a
5
− (k − 1) (from (4.2))
> k
1
+ k
3
+ a
2
+ 2a
4
(since k ≤ a
5
)
≥ a
2
.
We have only to show the fifth inequality, and this is immediate using (4.4) and our
hypothesis. This completes the proof of our sub claim.
We return to our monomial M = a
k
1
+ℓ
1
b
k
2
+ℓ
2
c
k
3
+ℓ
3
d
k
4
+ℓ
4
. By our subclaim, M ∈
(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) if and only if k
2
+ k
4
≥ a
5
− (k − 1). In order for M to have a chance
to be a minimal generator of (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
), we have to choose k
1
, k
2
, k
3
, k
4
as small
as possible. This means that we may assume that M has the form
a
ℓ
1
b
ℓ
2
+k
2
c
ℓ
3
d
ℓ
4
+k
4
where k
2
+k
4
= a
5
−(k −1). We have to show that this is in fact a multiple of an element,
N, of I(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
). We assert that the following is the desired element:
N
′
= a
ℓ
1
−k
2
b
ℓ
2
+k
2
c
ℓ
3
−k
4
d
ℓ
4
+k
4
.
To see this, consider the matrix (4.1). Recall that the monomial N = a
ℓ
1
b
ℓ
2
c
ℓ
3
d
ℓ
4
lies
in the removed Northwest corner (see diagram (4.3)), on the k-th diagonal. Each step
South represents a decrease of ℓ
1
by 1 and an increase of ℓ
2
by 1, while each step East
represents a decrease of ℓ
3
by 1 a nd an increase of ℓ
4
by 1. The monomial N
′
(accepting
temporarily that the exponents are non-negative) represents a move from N of k
2
steps
South and k
4
steps East. Each such step moves one to the next diagonal. But k
2
+ k
4
=
12 J. MIGLIORE, U. NAGEL
a
5
− (k − 1), so the result of the k
2
+ k
4
steps shows that N
′
lies on the border of (4.3),
on the diagonal in the Northwest corner. Hence N
′
∈ I(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) and so also
M ∈ I(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
). This completes the proof of the claim. In particular, we have
shown that (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) is generated in degree a
1
+ a
6
, and the value of β
1
is clear
from the construction (see the diagram (4.3)).
Having found the minimal generators of our S-minimal curve we can use the theory
of cellular resolutions as developed in [2]. Hence, we will first define an appropriate
regular cell complex X. Together with a choice of an incidence function, the cell complex
determines a complex F
X
of free R-modules. Finally, we will show that it is acyclic and
resolves the ideal of our curve.
As above we begin by describing the cell complex X
0
for the curve (a
1
, 0, 0, 0, 0, a
6
). Its
geometric realization will be a rectangle in the (i, j)-plane. To this end we will identify
the integer point (i, j) with the monomial a
j
b
a
1
−j
c
a
6
−i
d
i
. The vertices of X
0
are all the
integer points (i, j) satisfying 0 ≤ i ≤ a
6
and 0 ≤ j ≤ a
1
. The edges of X
0
are the
relative interiors of all “horizontal” line segments with endpoints (i − 1, j) and (i, j) and
all “vertical” line segments with endpoints (i, j − 1) and (i, j) where all the endpoints are
vertices of X
0
. The 2-dimensional faces of X
0
are the relative interiors of all squares with
vertices (i − 1, j − 1), (i, j − 1), (i, j), (i − 1, j), all being vertices of X
0
. Together with
the empty set, X
0
is clearly a finite regular cell complex (cf., e.g., [6], Section 6 .2 for the
definition). The empty set has dimension −1.
The cell complex X := X(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) of the curve (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) is a sub-
complex of X
0
obtained by deleting corners o f X
0
. More precisely, the vertices of X are
the points (i, j) satisfying
(4.6)
a
3
≤ i + j ≤ a
1
+ a
6
− a
4
a
5
− a
1
≤ i − j ≤ a
1
− a
2
The edges and 2-dimensional faces of X a r e by definition the corresponding faces of X
0
whose edges all belong t o X.
(4.7)
a
1
@
@
@
@
@
a
3
a
3
a
5
a
5
a
4
a
2
a
2
a
4
|
{z }
a
6
In the above diagram, the full rectangle indicates the cell complex X
0
, while the b old
face segments are the boundary of the cell complex X. Again it is clear that X is a finite
regular cell complex. For a face F of X we set m
F
the least common multiple of the
TETRAHEDRAL CURVES 13
vertices of F . The exponent vector of the monomial m
F
in Z
4
is called the degree of the
face F .
Now we fix an incidence function ε(F, F
′
) on pairs of faces of X (see fo r instance [2]).
Then the cellular complex F
X
is the Z-graded complex
F
X
: 0 −−−→ F
2
ϕ
2
−−−→ F
1
ϕ
1
−−−→ F
0
ϕ
0
−−−→ 0
where
F
i
=
M
F ∈X,dim F =i
R(− deg m
F
) =
M
F ∈X,dim F =i
Re
F
and the different ia l is given by
(4.8) ϕ
i
(F ) =
X
F
′
∈X,dim F
′
=i−1
ε(F, F
′
)
m
F
m
F
′
e
F
.
By the construction of this complex, the image of ϕ
0
is the ideal (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
),
thanks to the computation above of the minimal generators. We want to show tha t F
X
provides a minimal free resolution of this ideal. To this end denote for b = (b
0
, . . . , b
3
) ∈ Z
4
by X
≤b
the sub-complex of X on the vertices of degree ≤ b. Here, we use the partial
ordering on Z
3
induced by comparing componentwise. It is easy to see that for all de-
grees b ∈ Z
4
the geometric realization of the cell complex X
≤b
is contractible, thus X
≤b
is acyclic. Hence, [2], Proposition 1.2, shows that F
X
is a free resolution of the ideal
(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
).
Next, we observe for every i-dimensional face F of X that m
F
has degree a
1
+ a
6
+ i.
It follows immediately that F
X
is a minimal free resolution. Moreover, it is easy to see
that X contains (a
1
+ 1)(a
6
+ 1) −
P
5
i=2
a
i
(a
i
+1)
2
vertices, 2a
1
a
6
+ a
1
+ a
6
−
P
5
i=2
a
i
(a
i
+ 1)
edges, and a
1
a
6
−
P
5
i=2
a
i
(a
i
+1)
2
2-dimensional faces. Hence the complex F
X
giving the free
resolution of C is of t he form as claimed.
Combining this result with the Auslander-Buchsbaum formula we get:
Corollary 4.3. An S-mi nimal tetrahedral curve is arithmetically Cohen-Macaulay if and
only if it is trivial.
Remark 4.4. (i) In [2] Bayer and Sturmfels introduce the hull resolution. They show
that the hull resolution of a monomial ideal is a free resolution which is not necessarily
minimal. The hull resolution contains as sub-complex the so-called Scarf complex which
is easy to compute. For g eneric monomial ideals the Scarf complex and the hull resolution
agree and provide a minimal free r esolution. However, the defining ideal of a tetrahedral
curve is typically not generic (in the sense of Bayer and Sturmfels). For example, the
ideal (1 , 0, 0, 0, 0, 1) corresponding to a pair of skew lines has a Scarf complex of length 2
while its minimal free resolution ha s length 3 (cf. [23], Example 3.4.20). Tappe has a lso
computed the hull resolution of the ideals (a
1
, 0, 0, 0, 0, a
6
) for a
1
, a
6
≤ 3. In these cases
the hull resolution does give the minimal free resolution. However, it is not clear (to the
authors) if this is true for all S-minimal curves.
Note that the computation of the hull resolution requires one to compute the convex
hull of as many points as the ideal has minimal generators. This is a ra t her non-trivial
task if the number of points is large. On the other ha nd our description of the minimal
14 J. MIGLIORE, U. NAGEL
free resolution of S-minimal curves uses a very simple cell complex which gives a more
direct approach.
(ii) After we had found the cell complex that determines the minimal free resolution of
an S-minimal curve we realized that Schwar tau [20] uses a similar description in order to
compute the minimal free resolution of any tetrahedral curve with a
2
= a
5
= 0. However,
much of the theory that we have used here was not available during the writing of [20],
and we found that proof to be less “clean” even in the restricted setting.
As another consequence of Theorem 4.2 we get some information about the Hartshorne-
Rao module of any tetrahedral curve. Recall that the Hartshorne-Rao module of a curve
C is the gra ded module
M(C) := ⊕
j∈Z
H
1
(I
C
(j)).
Corollary 4.5. The K-dual of the Hartshorne-Rao module of a tetrahedral curve is gen-
erated in one degree.
Proo f . Since the Hartshorne-Rao module is up to degree shift invariant in an even liaison
class, it suffices to show the claim for a non-trivial S-minimal curve C. By local duality
we have the following graded isomorphism
M(C)
∨
∼
=
Ext
3
(R/I
C
, R)(−4).
The latter module can be computed from the minimal free resolution of I
C
. Since it is
linear the claim follows.
Remark 4.6. (i) The conclusion of the last result is in general not true for the
Hartshorne-Rao module itself. Indeed, the Hartshorne-Rao module of the curve
(5, 2, 2, 1, 1, 5) has minimal generators in two different degrees, namely 4 and 5.
(ii) The curve in (i) also shows that in general we cannot link a t etrahedral curve in an
odd number of steps to a tetrahedral curve.
Fro m a cohomological po int of view the simplest curves that are not arithmetically
Cohen-Macaulay are the arithmetically Buchsbaum curves. The tetrahedral curves among
them have special properties.
Corollary 4.7. A tetrahedra l curve is arithmetically Buchsbaum if and only if its
Hartshorne-Rao module satisfies
M(C)
∼
=
K
m
(t) for some integers m, t where m ≥ 0.
Proo f . This follows immediately by Corollary 4 .5.
Remark 4.8. The fact that such arithmetically Buchsbaum curves exist was known
already to Schwartau, who constructed them with liaison addition. The fact that they
are the only arithmetically Buchsbaum curves among the tetrahedral curves is new.
5. Minimality in the Even Liaison Class, and Applications
Now we want to show that for tetrahedral curves the concept of S-minimality and
minimality in its even liaison class agree. This is the second main result of this paper.
Theorem 5.1. Let C be a tetrahedral curve wh i ch is not arithmetically Coh e n-Macaulay.
Then C is S-minimal if and only if it is minimal in i ts even liaison class.
TETRAHEDRAL CURVES 15
Proo f . It suffices to show that every S-minimal curve is minimal in its even liaison class.
Denote by s(C) the initial degree of C,
s(C) := min{j ∈ Z | [I
C
]
j
6= 0},
and by e(C) its index of speciality,
e(C) := max{j ∈ Z | H
2
(I
C
(j)) 6= 0}.
A result of Lazarsfeld and Rao in [10] says that C is minimal if
s(C) ≥ e(C) + 4.
We will show that this criterion applies to our tetrahedral curves. We may again assume
that a
6
= max{a
i
} > 0. The minimal free resolution of C is
0 → R
β
3
(−a
1
− a
6
− 2)
φ
2
→ R
β
2
(−a
1
− a
6
− 1) → R
β
1
(−a
1
− a
6
) → I
C
→ 0
ց ր
K
ր ց
0 0
where K splits the resolution into two short exact sequences. We know that s(C) = a
1
+ a
6
,
so we have to show that h
2
(I
C
(a
1
+ a
6
− 3 )) = 0. (See Remark 5.5 (ii).) Letting K be the
sheafification of K, it is enough to show that h
3
(K(a
1
+ a
6
− 3)) = 0. The leftmost short
exact sequence gives
0 → H
2
(K(a
1
+ a
6
− 3)) → H
3
(O
β
3
P
3
(−5)) → H
3
(O
β
2
P
3
(−4)) → H
3
(K(a
1
+ a
6
− 3)) → 0.
Since the rightmost short exact sequence gives h
1
(I
C
(t)) = h
2
(K(t)) for all t, the above
long exact sequence gives
(5.1) h
3
(K(a
1
+ a
6
− 3)) = β
2
− 4β
3
+ h
1
(I
C
(a
1
+ a
6
− 3)).
Now, recall from [19], Theorem 2.5, that if C is a curve and if M(C) has minimal free
resolution
0 → L
4
σ
4
−→ L
3
→ L
2
→ L
1
→ L
0
→ M(C) → 0
then I
C
has minimal free resolution
0 → L
4
(σ
4
,0)
−→ L
3
⊕ F → F
1
→ I
C
→ 0.
In our case we know the minimal free resolution of I
C
. We first claim that F = 0, i.e.
the last two f r ee modules (and the map between them) in the minimal free resolution
of I
C
exactly coincide with the corresponding ones for M(C). In the proof of Theorem
4.2 we have described the maps in the minimal free resolution of C. Denote by M the
matrix describing the map φ
2
after choosing canonical bases for the free modules F
i
, where
we think of the columns of M as second syzygies. We want to show that M does not
have a row of zeros. But if it did, this means that there is a first syzygy which does
not “contribute” to any second syzygy. This gives a contradiction since the first syzygies
correspond to edges of the cell complex X, but every edge of X is in the boundary of a
facet of X that corresponds to a second syzygy. (This is a consequence of the inequalities
(4.2).)
16 J. MIGLIORE, U. NAGEL
Thus we know the end of the minimal free resolution fo r M(C) has the form
0 → R
β
3
(−a
1
− a
6
− 2)
φ
2
−→ R
β
2
(−a
1
− a
6
− 1) → . . . ,
and as a result the minimal f r ee resolution for the dual module (over the field K) M(C)
∨
has the form
· · · → F → R
β
2
(a
1
+ a
6
− 3)
φ
∨
2
−→ R
β
3
(a
1
+ a
6
− 2) → M(C)
∨
→ 0.
It is a basic property of minimal free resolutions that F has no summand of the form
R(a
1
+ a
6
− 3). Hence dim M(C)
∨
−a
1
−a
6
+3
= 4β
3
− β
2
, so (5.1) gives h
3
(K(a
1
+ a
6
− 3)) = 0
as desired.
Corollary 5.2. Assume that a
6
= max{a
1
, . . . , a
6
}. Then the degree of an S-minimal
curve of type (a
1
, . . . , a
6
) is
P
6
i=1
a
i
+1
2
, and the arithm e tic genus is
"
6
X
i=1
a
i
+ 1
2
#
(a
1
+ a
6
− 1) + 1 −
a
1
+ a
6
+ 2
3
.
Proo f . The degree statement is obvious. As f or the arithmetic genus, let us denote it by
g. It follows easily from the calculations in the proof of Theorem 5.1 that
h
0
(I
C
(a
1
+ a
6
− 1)) = h
1
(I
C
(a
1
+ a
6
− 1)) = h
2
(I
C
(a
1
+ a
2
− 1)) = 0.
Fro m the cohomology of the exact sequence
0 → I
C
→ O
P
3
→ O
C
→ 0
(twisted by a
1
+ a
6
− 1) it follows that
h
0
(O
C
(a
1
+ a
6
− 1)) =
a
1
+ a
6
+ 2
3
.
We also have h
1
(O
C
(t)) = h
2
(I
C
(t)) for all t. But the Riemann-Roch theorem gives that
h
0
(O
C
(a
1
+ a
6
− 1)) = (deg C)(a
1
+ a
6
− 1) − g + 1 + h
1
(O
C
(a
1
+ a
6
− 1)).
Combining the above calculations gives the r esult.
The following theorem can be viewed as a clarification of Schwartau’s theorem (see
Theorem 2.4 and Remark 2.5) and its proof is new and uses the methods of our paper.
The main t ool is Lemma 3.8, but we have put it after Theorem 5.1 because we use the
fact that an S-minimal curve is not ACM, and this fo llows from the fa ct that a curve that
is minimal in its even liaison class is not ACM.
Theorem 5.3 (invariant Schwartau). The ideal
(a, b)
a
1
∩ (a, c)
0
∩ (a, d)
a
3
∩ (b, c)
a
4
∩ (b, d)
0
∩ (c, d)
a
6
defines an arithmetically Cohen-Macaulay curve in P
3
if and only if
Case 1
. a
1
, a
3
, a
4
, a
6
> 0: a
1
+ a
6
= a
3
+ a
4
+ ǫ, where ǫ ∈ {−1, 0, 1}.
Case 2
. Exactly one of a
1
, a
3
, a
4
, a
6
is zero, say a
i
: a
7−i
+ 1 ≥ the sum of the weights of
the l i nes meeting the line l
i
.
TETRAHEDRAL CURVES 17
Case 3. At least two of a
1
, a
3
, a
4
, a
6
are zero: the curve is connected .
Proo f . For convenience we will call a curve of the form given in Theorem 5.3 a Schwartau
curve. If C is a Schwartau curve then its components can be represented by a square:
a
1
a
3
a
4
a
6
If three of the integers a
i
are zero then the ideal is a power of a complete intersection, and
is automatically ACM and connected. Suppose that two of the integers a
i
are zero. If C
is ACM then it is a general fact that it must be connected. Conversely, suppose that C is
connected. This means that the two non-zero values of a
i
represent a djacent sides of the
square. Hence the ideal involves only three variables. The fourth variable is thus a non
zero-divisor, and reducing modulo this variable gives the saturated ideal of a zeroscheme
in P
2
. Hence in this case the Schwartau curve is ACM. This proves Case 3.
Now suppose that one of t he integers is zero, say a
i
. (Note that Schwartau took a
3
= 0.)
As long as C is not S-minimal, we can reduce any maximal facet, by Lemma 3.8. But
with a
i
= 0, there remain only two facets, whose weights are the sums of two consecutive
sides of the square. (For example, if a
3
= 0 then the weights of the facets are a
4
+ a
6
and
a
1
+ a
4
.)
If C is ACM then no reduction will ever be S-minimal. We repeatedly reduce the facet
of maximal weight until we obtain another zero for the weight of an edge, and the fact
that C is ACM means that after obtaining this zero, the remaining two edges must be
connected (by Case 3 above). This means that the “middle” of the three non-zero edges,
a
7−i
, must not be the first to reach 0 unless there is a tie. But this middle edge is involved
in every reduction! Note that since we are always reducing the facet of maximal weight,
we event ually reach the point where the weights of the non-middle edges differ by at most
1. It follows that a
7−i
+ 1 ≥ the sum of the weights of the two “non-middle” edges, as
claimed.
Conversely, suppose that a
7−i
+ 1 ≥ the sum of the weights of the two “non-middle”
edges. Then clearly a
7−i
is the edge of maximal weight. The condition (b) of Lemma 3.8
is trivial to check, so C is not minimal and we can reduce the facet of maximal weight.
In doing so, bo t h sides of the inequality in Case 2 are reduced by 1, so the inequality still
holds for the new curve. Aga in, reducing the facet o f maximal weight each time eventually
leads to a balance in the two non-middle edges, and the given inequality guarantees that
the curve resulting when a second 0 is obtained, will be connected, hence ACM. So C is
ACM.
We now consider the case where none of the a
i
are 0. Again without loss of gener-
ality assume that a
6
= max{a
1
, a
3
, a
4
, a
6
}. One can check that then w = max{a
3
+ a
6
,
a
4
+a
6
}. It follows that when we reduce a facet of maximal weight (if it is po ssible), we si-
multaneously reduce max{a
1
, a
6
} and max{a
3
, a
4
} by 1. In particular, we simultaneously
reduce (a
1
+ a
6
) by 1 and (a
3
+ a
4
) by 1.
18 J. MIGLIORE, U. NAGEL
If C is ACM then it is not S-minimal, so by Lemma 3.8 we have
a
1
+ a
6
≤ max{a
3
+ a
6
, a
4
+ a
6
}, i.e. a
1
≤ max{a
3
, a
4
}.
We conclude that if C is ACM then the largest two weights are on consecutive edges. That
is, the facet of maximal weight is one o f the two f acets involving a
6
, and we can reduce
that facet. Since C is ACM, we will never reach an S-minimal curve in this process, and
the result of performing a reduction will a gain be an ACM curve. Hence after each step
in the reduction, the two edges of largest weight are consecutive and each can be reduced
by 1 by a new reduction.
Suppose that we have reached the point where the application of Proposition 3.1 reduces
an a
i
to zero. If two edges are reduced to 0 simultaneously, the remaining two edges a r e
consecutive and have weight 1 each (at this point). So because of our procedure, it follows
that a
1
+ a
6
= a
3
+ a
4
.
If one edge is reduced to 0, suppose that it is supported on ℓ
i
. Then at this point ℓ
7−i
has weight 1, and the resulting curve is ACM. But ℓ
7−i
is then the middle edge, so by Case
2 we have the other two edges having weight 1 each. It follows that a
1
+ a
6
= a
3
+ a
4
+ ǫ
as claimed.
Conversely, assume that a
1
+ a
6
= a
3
+ a
4
+ ǫ for ǫ = −1, 0, 1. We want to show
that a reduction can be performed, i.e. C is not minimal. Without loss of generality we
still assume that a
6
is the la rgest weight. We claim that a
1
≤ max{a
3
, a
4
}. Indeed, if
a
6
≥ a
1
> max{a
3
, a
4
} then a
1
+ a
6
≥ a
3
+ a
4
+ 2, contradicting the hypothesis. But then
if w is the maximal weight of a facet, we have w = a
6
+ max{a
3
, a
4
} and a
1
+ a
6
≤ w.
Therefore, by Lemma 3.8, a reduction can be performed.
Hence our numerical assumption guarantees that at least as long as the four integers
are positive, we can always perform a reduction, reducing the lar ger of a
1
and a
6
by 1
and the larger of a
3
and a
4
by 1. Suppose that a
3
is the first to be reduced to zero. Then
at that point a
4
= 1 and a
1
+ a
6
is either 1 or 2. Either way, we are in the previously
studied cases and C is ACM.
We can also characterize the S-minimal arithmetically Buchsbaum curves.
Corollary 5.4. Let C be an S-minimal tetrahedra l curve defined by (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
).
Then C is arithmetically Buchsbaum and not arithmetically Cohen-Macaulay if and only
if there are integers i 6= j in {1, 2, 3} such that
a
i
+ 1 = a
j
= a
7−j
= a
7−i
+ 1
and the two remaining weights are zero.
Proo f . According to Corollary 4.3 we may assume that a
6
= max{a
i
} > 0. We will
use the notation of Theorem 4.2 and its proof . Since C is minimal, a result of Martin-
Deschamps and Perrin says that the transpose of the matrix describing the map ϕ
2
is a
minimal presentation matrix of Ext
3
(R/I, R). (See [19] Theorem 2.5 and [11] Proposition
IV.4.4.) We also saw this directly, in the case of tetrahedral curves, in the proof above.
By Corollary 4.7, C is arithmetically Buchsba um if a nd only if the latter module is a
(shifted) direct sum of copies o f K. Using the cell complex X that governs the minimal
free resolution of C, this is equivalent to the f act that every edge of X belongs to the
boundary of exactly one facet. Now the claim follows from the description of X. Indeed,
TETRAHEDRAL CURVES 19
after removing diagonals from the corners of (4.1 ) and translating to the notation of the
cell complex X, we must be left with a diagonal of facets and no additional edges, and
this can only happen if we only remove diagonals from two opposite corners of (4.1) in
the prescribed way.
Remark 5.5. (i) The arguments in the proof of Theorem 5.1, coupled with a recent result
of Strano [22], Theorem 1, show the following fact:
If a non-arithmetically Cohen-Macaulay curve C ⊂ P
3
has a linear resolution then it is
minimal in its even liaison class if and only of s(C) ≥ e(C) + 4.
(ii) There are tetrahedral curves that have a linear resolution, but are not minimal in
their even liaison class. An example is given by the curve with weights ( 5 , 1, 3, 2, 2, 5). This
shows that our task in the proof of Theorem 5.1, of showing that h
2
(I
C
(a
1
+ a
6
− 3)) = 0,
cannot be shown merely by invoking the linearity of the resolution. Indeed, this only
guarantees h
2
(I
C
(a
1
+ a
6
− 2)) = 0.
(iii) Sometimes a non-trivial S-minimal curve is the unique minimal curve in its even
liaison class. This is true if s(C) ≥ e(C) + 5 (thanks to [10]). Examples of such curves
have weights (m, 0, 0, 0 , 0, k) where m, k ≥ 2.
6. Unobstructedness of some curves
We will show that some of the minimal tetrahedral curves correspond to smooth points
of a component of the corresponding Hilbert scheme that has the “expected” dimension.
One starting point is the following result whose proof is an adaptation of Dolcetti’s
method in [7]. Note that if d
1
is the degree of the first non-zero component of a g r aded
module, M, of finite length and d
2
is the degree of the last non-zero component of M,
then the diameter of M is d
2
− d
1
+ 1.
Proposition 6.1. Let C ⊂ P
3
be a curve with a linear resolution. If the diameter of its
Hartshorne-Rao module is at most two then its normal sheaf N
C
satisfies
H
1
(N
C
) = 0.
Therefore, the corresponding component of the Hilbert scheme is generically smooth of
dimension 4 · deg C.
Proo f . By assumption the minimal free resolution of C is of the form
0 → R
β
3
(−s − 2) → R
β
2
(−s − 1) → R
β
1
(−s) → I
C
→ 0.
It implies
H
1
(I
C
(s − 2)) 6= 0 and H
1
(I
C
(j)) = 0 if j ≥ s − 1
as well as
H
2
(I
C
(j)) = 0 for all j ≥ s − 2.
Thus, the assumption on the diameter provides
H
1
(I
C
(j)) = 0 for all j ≤ s − 4.
20 J. MIGLIORE, U. NAGEL
Sheafifying the resolution above we define the vector bundle E as follows
0 → O
β
3
P
3
(−s − 2) → O
β
2
P
3
(−s − 1) −→ O
β
1
P
3
(−s) → I
C
→ 0
ց ր
E
ր ց
0 0
Thus, we have
H
1
∗
(E) = 0 and H
2
∗
(E) = H
1
∗
(I
C
).
Furthermore, dualizing the exact sequence on the left-hand side and then tensoring by E
we get the exact sequence
0 → E
∗
⊗ E → E
β
2
(s + 1) → E
β
3
(s + 2) → 0.
Taking cohomology we obtain
H
2
(E
∗
⊗ E) = 0.
Now, the cohomology of the exact sequence
0 → E
∗
⊗ E → (E
∗
)
β
1
(−s) → E
∗
⊗ I
C
→ 0
provides
(H
1
(E
∗
(−s)))
β
1
→ H
1
(E
∗
⊗ I
C
) → H
2
(E
∗
⊗ E) = 0.
Since we have by duality
H
1
(E
∗
(−s))
∼
=
[Ext(R/I
C
, R)]
−s
∼
=
H
1
(I
C
(s − 4)) = 0
we conclude
H
1
(E
∗
⊗ I
C
) = 0.
The sequence defining E also gives the exact sequence
Ext
1
(E, I
C
) → Ext
2
(I
C
, I
C
) → Ext
2
(O
β
1
P
3
(−s), I
C
).
Using
Ext
2
(O
β
1
P
3
(−s), I
C
)
∼
=
(H
2
(I
C
(s)))
β
1
= 0
and
Ext
1
(E, I
C
)
∼
=
H
1
(E
∗
⊗ I
C
) = 0
we get
0 = Ext
2
(I
C
, I
C
)
∼
=
H
1
(N
C
),
as claimed.
Since minimal tetrahedral curves have a linear resolution we would like t o know the
ones whose Hartshorne-Rao module has diameter at most two. The curves with diameter
one have already been characterized in Corollary 5.4. The diameter two case is described
below:
Lemma 6.2. Let C be a minimal tetrahedral curve. Then its Hartshorne-Rao module has
diameter two if and only if C is isomorphic to one of the curves
(k, k − 1, 0, 0, k − 1, k + 1) where k ≥ 1, or
(k, k − 2, 0, 0, k − 1, k) where k ≥ 2.
TETRAHEDRAL CURVES 21
Proo f . Let C be the minimal curve defined by the tuple (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
). Without
loss o f generality assume that a
6
is the largest entry. We will use the description of the
minimal free resolution of C given in Theorem 4.2. Denote by ψ the dual of the last map
in this resolution, i.e. ψ = ϕ
∗
2
. Thus, we have the minimal presentat io n
−−−→ R
β
3
(a
1
+ a
6
+ 1)
ψ
−−−→ R
β
2
(a
1
+ a
6
+ 2) −−−→ Ext
3
(R/I
C
, R) −−−→ 0
Since Ext
3
(R/I
C
, R) is the K-dual of the Hartshorne-Rao module M
C
of C, the diameter
of the latter is at most two if and only if
(6.1) [Ext
3
(R/I
C
, R)]
−a
1
−a
6
= 0.
Denote by X := X(a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) the cell complex of the curve C. Recall from
the proof of Theorem 4.2 that the facets of X correspond to minimal generators of
Ext
3
(R/I
C
, R) and that im ψ has a system of minimal generators consisting of bino-
mials and monomials (cf. formula (4.8)). These generators correspond to the edges of X.
Below, we will always refer to this system of generators. Moreover, we will use Diagram
(4.7).
We begin with deriving necessary conditions for Condition (6.1) being true. In order
to rule out curves we show two claims.
Claim 1: If X contains “3 facets in a row” then the diameter of M
C
is greater than
two.
Here the assumption means that X contains a subcomplex that is isomorphic to
X(3, 0, 0, 0, 0, 1) (a “vertical row”) or X(1, 0, 0, 0, 0, 3) (a “horizontal row”).
Now, assume that X contains 3 facets in a horizontal row. Denote by e
1
, e
2
, e
3
these
facets from left to right. Then, the only minimal generators of im ψ involving ce
1
, ce
2
, de
2
,
or de
3
are (up to sign)
−ce
1
+ de
2
and − ce
2
+ de
3
.
It follows easily that all the monomials c
2
e
1
, cde
2
, d
2
e
3
do not belong to im ψ. Thus,
condition (6.1) is not satisfied.
If X contains 3 facets in a vertical row we argue similarly and Claim 1 is shown.
Claim 2: If X contains a “square of 4 facets” then the diameter of M
C
is greater than
two.
More precisely, the assumption means that X contains a subcomplex that is isomor-
phic to X(2, 0, 0, 0, 0, 2). Enumerate its facets counterclockwise by e
1
, . . . , e
4
beginning
with e
1
in the “Southwest” corner. Then, t he only minimal generators of im ψ involving
be
1
, ce
1
, be
2
, de
2
, ae
3
, de
3
, ae
4
, or ce
4
are (up to sign)
− ce
1
+ de
2
, −be
2
+ ae
3
− ce
4
+ de
3
, −be
1
+ ae
4
.
Hence, all the monomials bce
1
, bde
2
, ade
3
, ace
4
are not in im ψ. Thus, condition (6.1) is
not satisfied and Claim 2 is established.
As the next step we look f or the minimal tetrahedral curves whose cell complex contains
neither 3 facets in a row nor a square of 4 facets. Let C be such a curve. Looking at the
22 J. MIGLIORE, U. NAGEL
boundaries of its cell complex X we get in conjunction with Corollary 3.5
1 ≤ a
6
− a
4
− a
5
≤ 2
1 ≤ a
6
− a
2
− a
3
≤ 2
1 ≤ a
1
− a
3
− a
5
≤ 2
1 ≤ a
1
− a
2
− a
4
≤ 2.
Now, we look at a horizontal or vertical boundary o f X a bit more carefully. If there are
true “cuts” on both ends then the row of X next to it has length at least 3, a contradiction.
Thus, we may assume without loss of generality that
a
3
= 0.
Next, we distinguish two cases.
Case 1: Assume a
2
= 0. Then we get a
6
≤ 2 and it is easy to see that C can be any
curve satisfying these conditions except t he curve defined by (2, 0, 0, 0, 0, 2).
Case 2: Assume a
2
> 0. Using the argument above for the Eastern boundary of X we
conclude
a
4
= 0.
We are left with two possibilities for the value of a
5
.
Case 2.1: Assume a
5
= a
6
− 1. Since a
5
< a
1
≤ a
6
we obtain a
1
= a
6
. We conclude
that either a
2
= a
6
− 1, i.e. C is an arithmetically Buchsbaum curve of type
(k, k − 1, 0, 0, k − 1, k),
or a
2
= a
6
− 2, i.e. C is of type
(k, k − 2, 0, 0, k − 1, k).
Case 2.2: Assume a
5
= a
6
− 2. Since the length of the row next to the Northern
boundary of X is at most 3 we get a
2
= a
1
− 1. It follows that either a
1
= a
6
, i.e. C is
isomorphic to the second kind of curve in Case 2.1, or a
1
= a
6
− 1, i.e. C is of type
(k, k − 1, 0, 0, k − 1, k + 1).
Summing up. we have shown that there are at most three types of minimal tetrahedral
curves satisfying Condition (6 .1 ) . Since we already know all the curves with diameter one
(by Corollary 4.7), it remains t o show that the diameter of the curves specified in the
statement is at most two.
Let C be the curve defined by
(k, k − 1, 0, 0, k − 1, k + 1).
Enumerate the facets of the cell complex X of C by e
1
, . . . , e
2k
such that e
1
is the facet in
the Southeast corner and the facets e
i
and e
i+1
are neighbours. This enumeration exists
and is uniquely determined due to the shape of X. We will show that
(6.2) (a, b, c, d)
2
· e
i
∈ im ψ for all i = 1, . . . , 2k.
First, we consider e
1
. Since (a, b, d) · e
1
∈ im ψ it suffices to show c
2
e
1
∈ im ψ. But this
follows easily using
c · (−ce
1
+ de
2
) ∈ im ψ
TETRAHEDRAL CURVES 23
and
ce
2
∈ im ψ.
Next, we consider e
2
. Since also ae
2
∈ im ψ it remains to show (b, d)
2
· e
2
∈ im ψ. Using
the minimal generators −ce
1
+ de
2
, be
1
, de
1
we see that
(bd, d
2
)e
2
∈ im ψ.
We also have
b
2
e
2
∈ im ψ
because −be
2
+ ae
3
, be
3
∈ im ψ.
Continuing similarly in this fashion we can show the relations (6.2) that imply Condition
(6.1). This completes the argument.
Corollary 6.3. Let C be a curve that is isomorphic to one of the curves
(k, k − 1, 0, 0, k − 1, k) where k ≥ 1,
(k, k − 1, 0, 0, k − 1, k + 1) where k ≥ 1, or
(k, k − 2, 0, 0, k − 1, k) where k ≥ 2.
Then C is unobstructed and the corresponding component of the Hilbert scheme is ge ner-
ically smooth and has dimension 4 · deg C.
Proo f . This follows by combining Proposition 6.1, Lemma 6.2, and Theorem 4.2.
Remark 6.4. It was proved in [4] that if C is a curve in P
3
(not necessarily tetrahedral)
with natural cohomology and if the diameter of M(C) is two, then C is unobstructed and its
Hilbert scheme has the expected dimension. “Natural cohomology” means that for any t,
no two of h
0
(I
C
(t)), h
1
(I
C
(t)) and h
2
(I
C
(t)) are non-zero. Note that minimal tetrahedral
curves whose Hartshorne-Rao modules have diameter two have natural cohomology.
Note also that our Proposition 6.1, while similar, is independent of that result. Fo r
example, a non-minimal tetrahedral curve with linear resolution (e.g. (6, 5, 1, 0, 4, 6)) does
not have natural cohomology, but is unobstructed by Proposition 6.1. On the other hand,
a minimal arithmetically Buchsbaum curve C with M(C) of diameter two does not have
a linear resolution (since the dual module would then be generated in more than one
degree), but it does have natural cohomology (cf. [3 ], for instance Corollary 2.5 with t = 2
and h = 0) and so is unobstructed.
We also remark that Mir´o-Roig has given sufficient conditions on the numerical char-
acter of an irred ucib l e arithmetically Buchsbaum curve C of maximal rank with M(C) of
diameter one or two, for C to be unobstructed (cf. [17]).
The condition on the diameter of the minimal tetrahedral curves in Propo sition 6.1 is
sufficient for unobstructedness, but not necessary. This is shown by the curves
(a
1
, 0, 0, 0, 0, a
6
) because their diameter is a
1
+ a
6
− 1, but all these curves are unob-
structed. This follows from the following result.
Proposition 6.5. The global sections of the normal sheaf of the curve C defined by
(a
1
, 0, 0, 0, 0, a
6
) are given by
H
0
∗
(N
C
)
∼
=
(R/(a, b))
a
1
(a
1
+1)
(1) ⊕ (R/(c, d))
a
6
(a
6
+1)
(1).
24 J. MIGLIORE, U. NAGEL
Before turning to the proof of this statement we record its announced consequence.
Corollary 6.6. Le t C ⊂ P
3
be the curve defined by (a
1
, 0, 0, 0, 0, a
6
). Then its normal
sheaf satisfies
H
1
(N
C
(−2)) = 0.
Therefore, the corresponding component of the Hilbert scheme is generically smooth of
dimension 4 · deg C.
Proo f . The claim easily follows by Propo sition 6.5 and duality.
The proof of Proposition 6.5 is based on the computation of the normal module of
infinitesimal neighbourhoods of a line. Recall that the normal module of the curve C is
N
C
= Hom(I
C
, R/I
C
).
Lemma 6.7. For any positive integer k there is a graded isomorphism
Hom((a, b)
k
, R/(a, b)
k
)
∼
=
(R/(a, b))
k(k+1)
(1).
Proo f . Put I = (a, b)
k
and A = R/I. The minimal free resolution of I is well-known
0 → R
k
(−k − 1) → R
k+1
(−k) → I → 0.
Dualizing it with respect to R provides the minimal fr ee resolution of t he canonical module
K
A
of A
0 → R → R
k+1
(k) → R
k
(k + 1) → K
A
(4) → 0.
Dualizing with respect to A we get the exact sequence
0 → Hom(I, A) → A
k+1
(k)
ψ
−→ A
k
(k + 1) → Ext
1
(I, A) → 0.
Thus, we see in particular that Ext
1
(I, A)
∼
=
K
A
⊗ A(4) which allows to compute its
Hilbert function. Using the last exact sequence above we obtain
rank
K
[Hom(I, A)]
j
=
0 if j ≤ −2
k(k + 1)(j + 2) if j ≥ −1
Now, let {e
1
, . . . , e
k+1
} be the canonical basis of the free A-module A
k+1
. Using (a, b) ·
(a, b)
k−1
= I we see that
(a, b)
k−1
e
i
⊂ [ker ψ]
−1
for all i = 1, . . . , k + 1.
Thus, the R-module M that is generated by
G := {a
j
b
k−1−j
e
i
| j = 0, . . . , k − 1, i = 1, . . . , k + 1}
is a submodule of Hom(I, A). It is not too difficult to see t hat the minimal generators of
M are annihilated by the ideal (a, b) and, thus, G is a basis of the free R/(a, b)-module M.
Comparing Hilbert functions we conclude t hat M = Hom(I, A) completing the proof.
We are ready for the proof of Proposition 6.5.
TETRAHEDRAL CURVES 25
Proo f of Proposition 6.5. Let C
1
, C
2
be the curves defined by (a, b)
a
1
and (c, d)
a
6
, respec-
tively. Using the exact sequence
0 → I
C
→ I
C
1
⊕ I
C
2
→ I
C
1
+ I
C
2
→ 0
we get
H
0
∗
(O
C
)
∼
=
H
0
∗
(O
C
1
) ⊕ H
0
∗
(O
C
2
).
Since H
0
∗
(N
C
)
∼
=
Hom(I
C
, H
0
∗
(O
C
)) it is not too difficult to see that the claim follows by
Lemma 6.7.
7. Remarks and problems
In this section we collect some observations that do not quite fit into earlier sections
of this paper, as well as some natural questions that arise from this work. The list of
questions is rather long, highlighting the richness of this line of inquiry. The authors plan
to continue investigating these questions.
Remark 7.1. It is natural to ask, among all tetrahedral curves, “how many” are minimal
in their even liaison class. That is, can we describe the density of minimal tetrahedral
curves among all tetrahedral curves? The first answer is “almost none,” since of course
every such even liaison class has infinitely many tetrahedral curves, but essentially one
minimal one (but see question 1. below). But para doxically, we can take a different point
of view that shows that there are more t han o ne might think.
We want to investigate how many minimal tetrahedral curves are in a finite set of
tetrahedral curves. To t his end it seems reasonable to fix the maximum entry of the 6-
tuple. After a change of coordinates we may assume that this entry is the last one. Then
we get:
Lemma 7.2. The number of minim al tetrahedral curves (a
1
, . . . , a
6
) such that a
6
=
max{a
1
, . . . , a
6
} is
N(a
6
) :=
a
6
X
a
1
=0
a
1
−1
X
a
2
=0
a
1
−1
X
a
5
=0
min{a
1
− a
5
, a
6
− a
2
} · min{a
1
− a
2
, a
6
− a
5
}.
Proo f . This is a consequence of Corollary 3.5. If (a
1
, . . . , a
6
) is minimal then we have
0 ≤ a
1
≤ a
6
and 0 ≤ a
2
, a
5
< a
1
. Moreover, having chosen a
1
, a
2
, a
5
, the only condition
for a
3
and a
4
is 0 ≤ a
3
< min{a
1
− a
5
, a
6
− a
2
} and 0 ≤ a
4
< min{a
1
− a
2
, a
6
− a
5
},
respectively. The claim f ollows.
The number of tetrahedral curves with fixed a
6
= max{a
1
, . . . , a
6
} is (a
6
+ 1)
5
. The
lemma shows that the number of minimal curves among them is also of order a
5
6
. To see
this it suffices to find a lower estimate of the number N(a
6
). Since both minima in the
26 J. MIGLIORE, U. NAGEL
formula in Lemma 7.2 are at least a
1
− min{a
2
, a
5
} we get
N(a
6
) ≥
a
6
X
a
1
=0
a
1
−1
X
a
2
=0
a
1
−1
X
a
5
=0
[a
1
− min{a
2
, a
5
}]
2
=
a
6
X
a
1
=0
a
1
−1
X
a
2
=0
"
a
2
−1
X
a
5
=0
(a
1
− a
5
)
2
+
a
1
−1
X
a
5
=a
2
(a
1
− a
2
)
2
#
=
a
6
X
a
1
=0
a
1
−1
X
a
2
=0
"
a
1
X
k=a
1
−a
2
+1
k
2
+ (a
1
− a
2
)
3
#
Now it is easy to see that this lower estimate of N(a
6
) is a polynomial in a
6
of order 5.
In other words, we get the somewhat surprising result: The probability to find a minimal
curve among the set of tetrahedral curves (a
1
, . . . , a
6
) such that a
6
= max{a
1
, . . . , a
6
} is
positive, even as a
6
→ ∞.
Remark 7.3. The results in this paper can be extended as follows. Let (F
1
, F
2
, F
3
, F
4
)
be a regular sequence. Define curves by I
C
1
= (F
1
, F
2
), I
C
2
= (F
1
, F
3
), . . . , I
C
6
= (F
3
, F
4
),
and let I
C
= I
C
1
∩ · · · ∩ I
C
6
. Taking (F
1
, F
2
, F
3
, F
4
) = (a, b, c, d) gives the context of this
paper. As before, we denote by (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
) the ideal I
C
. Note that no two of the
curves C
i
can have a common component.
Many of our results extend to this situation. The key observation is that we still have
F
i
· (F
i
, F
j
)
n−1
+ (F
n
j
) = (F
i
, F
j
)
n
,
as we noted at the beginning of the proof of Proposition 3.1. Then Proposition 3.1 and
Corollary 3.5 (a nd indeed virtually all of section 3) are still true, as stated, in this more
general context. Turning to Section 4, Theorem 4.2 is clearly not true as stated. If the F
i
all have the same degree, it is fairly easy to modify the statement, and in fact the linearity
of the resolution is preserved. If the F
i
have different degrees, it can still be modified,
but the resolution is no longer linear. The other results are less obviously generalizable
to this context.
We end with some open questions.
Question 7.4. (1) Fix an even liaison class containing tetrahedral curves. We know
that among the minimal curves there is at least one that is tetrahedral. When is a
minimal tetrahedral curve the unique minimal curve in the even liaison class (see
Remark 5.5, (iii))? Note tha t sometimes two curves that are projectively equiva-
lent are linked, and sometimes they are not!
(2) Is it possible to chara cterize t he tetrahedral curves with linear resolutions? Recall
that this set of curves contains more than just the minimal tetrahedral curves
(Remark 5.5 (ii)).
(3) We have seen that several of the minimal tetrahedral curves are unobstructed.
Computer experiments show that there are more of them than the ones described
above. Are all minimal tetrahedral curves unobstructed? Are all tetrahedral
TETRAHEDRAL CURVES 27
curves with linear resolution unobstructed? Are all tetrahedral curves unob-
structed?
(4) How “dense” are our minimal curves that give rise to nice compo nents of the
Hilbert scheme H
d,g
, i.e. components that are generically smooth of dimension 4d?
In other words: How big are the “gaps” in the sets of pairs (d, g) such we cannot
find such a minimal curve with that degree and genus?
(5) Can the tetrahedral curves in P
3
that are arithmetically Cohen-Macaulay be iden-
tified by explicitly giving the 6-tuples (as Schwartau does for 4-tuples)?
(6) Is there a combinatorial description of the minimal f r ee r esolution of the deficiency
module of a tetrahedral curve? Can we at least express the dimensions of the com-
ponents using the entries of the 6-tuple?
(7) Can the same kind of program be carried out in higher projective space? Now the
new question of local Cohen-Macaulayness arises. This includes codimension two
cases and higher codimension cases. The former can still rely on complete inter-
section liaison techniques, but the latter will require Gorenstein liaison techniques.
(8) On a computer algebra program such as macaulay [1] one can make experiments,
and one notices a great deal of structure in the Betti diagram. In particular, non-
minimal tetrahedral curves have linear strands in the resolution. How do these
arise? Certainly they come from the Betti diagram of the minimal tetrahedral
curve using basic double links, but this should be explained in a clearer way. Can
one relate the “degree of non-minimality” to the number of linear strands? The
result of [16] Corollary 4.5 should be useful.
(9) Is it possible to use similar techniques to study more general monomial ideals?
Appendix: The algorithm to find S-minimal curves
Below we record a crude MAPLE implementation of Algorithm 3.9. It is a MAPLE
work sheet that can be downloaded at
http://www.ms.uky.edu/∼uwenagel/.
> # algorithm to compute the S-minimal curve of a tetrahedral curve
# Input: weight vector of the given curve
> smin := proc(aa,bb,c,d,e,f)
local a, b, m, i, j, k, s, t, w, W, x, y, z, output1, output2;
a := [aa,bb,c,d,e,f];
b:= a; s := 0;
28 J. MIGLIORE, U. NAGEL
if (s > -1) then do
# 1. computation of the maximal weight m of a line
m := a[1]; j := 1;
for i from 2 to 6 do
if (a[i] > m) then m := a[i];
j := i;
end if;
od;
# 2. avoid unnecessary computations
if (m = 0) then
output1 := matrix(1,6,[[‘ ‘,‘Minimal curve to‘, b, ‘is‘, a,‘ ‘]]):
output2:= matrix(1,3,[[‘It is obtained after‘, s,‘ reduction(s).‘]]):
print(output1); print(output2);
return(a);
end if;
# 3. computation of facet of maximal weight W
x := a[1] + a[2] + a[3];
y := a[1] + a[4] + a[5];
z := a[2] + a[4] + a[6];
t := a[3] + a[5] + a[6];
w := [x, y, z, t];
W:= w[1]; k := 1;
for i from 2 to 4 do
if (w[i] > W) then W := w[i];
k := i;
end if;
od;
# 4. test if curve is S-minimal
if (a[j] + a[7-j] > W) then
output1 := matrix(1,6,[[‘ ‘,‘Minimal curve to‘, b, ‘is‘, a,‘ ‘]]):
output2:= matrix(1,3,[[‘It is obtained after‘, s,‘ reduction(s).‘]]):
print(output1); print(output2);
return(a);
end if;
# 5. reduction of the non-minimal curve
TETRAHEDRAL CURVES 29
s := s+1;
if (k = 1) then a[1] := max(0, a[1] - 1);
a[2] := max(0, a[2] - 1);
a[3] := max(0, a[3] - 1);
end if;
if (k = 2) then a[1] := max(0, a[1] - 1);
a[4] := max(0, a[4] - 1);
a[5] := max(0, a[5] - 1);
end if;
if (k = 3) then a[2] := max(0, a[2] - 1);
a[4] := max(0, a[4] - 1);
a[6] := max(0, a[6] - 1);
end if;
if (k = 4) then a[3] := max(0, a[3] - 1);
a[5] := max(0, a[5] - 1);
a[6] := max(0, a[6] - 1);
end if;
end do;
end if;
end:
In order to use the procedure above in a different work sheet one could write it into a
text file named, f or example, proc.txt. Then it can be used as follows.
> read ‘C:\\suitable path\\proc.txt‘;
> smin(5,1,3,2,2,5):
[ , Minimal curve to , [5, 1, 3, 2, 2, 5] , is ,
[5, 1, 2, 2, 1, 4] , ]
30 J. MIGLIORE, U. NAGEL
[It is obtained after 1 reduction(s).]
> smin(6,0,8,1,0,4):
[ , Minimal curve to , [6, 0, 8, 1, 0, 4] , is ,
[0, 0, 0, 0, 0, 0] , ]
[It is obtained after 10 reduction(s).]
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Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
E-mail address: Juan.C.Migliore.1@nd.edu
Department of Mathematics, University of Kentucky, 715 Patterson Office Tower,
Lexington, KY 40506-0027, USA
E-mail address: uwenagel@ms.uky.edu
