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Algebraic geometry; a celebration of Emma Previato's 65th birthday
Abstract
These two volumes celebrate and honor Emma Previato, on the occasion of her 65th birthday. The present volume consists of 16 articles in algebraic geometry and its surrounding fields, emphasizing the connections to integrable systems which are so central to Emma's work. The companion volume focuses on Emma's interests within integrable systems. The articles were contributed by Emma's coauthors, colleagues, students, and other researchers who have been influenced by Emma's work over the years. They present a very attractive mix of expository articles, historical surveys and cutting edge research.
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A recent generalization of the "Kleinian sigma function" involves the choice of a point P of a Riemann surface X, namely a "pointed curve" . This paper concludes our explicit calculation of the sigma function for curves cyclic trigonal at P. We exhibit the Riemann constant for a Weierstrass semigroup at P with minimal set of generators , , equivalently, non-symmetric, we construct a basis of and a fundamental 2-differential on , we give the order of vanishing for sigma on Wirtinger strata of the Jacobian of X, and a solution to the Jacobi inversion problem.
Don Zagier defined a “Rankin-Cohen algebra”, motivated by the study of differential operators that send modular forms to modular forms. We devised an algorithm that computes the result of the differentiation given by the modular forms that correspond to higher-order Wronskians over Klein’s quartic curve, which are modular forms of arbitrarily high degree canonically attached to the curve; this tool is potentially useful for finding commutative rings of differential operators.
The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing
isometry and isoenergy conditions on a moduli space of plane loops. The
conditions are compared to the constraints that define Euler's elastica.
Moreover, the conditions are shown to be constraints on the curvature and other
invariants of the loops which appear as coefficients of the generating function
for the Faber polynomials.
A cyclic trigonal curve of genus three is a Galois cover of
, therefore can be written as a smooth plane curve with equation
. Following Weierstrass
for the hyperelliptic case, we define an ``'' function for this
curve and , c=0,1,2, for each one of three particular
covers of the Jacobian of the curve, and r=1,2,3,4 for a finite branchpoint
. This generalization of the Jacobi , ,
functions satisfies the relation: which generalizes
. We also show that this can be viewed as
a special case of the Frobenius theta identity.
We explore some of the interplay between Brill-Noether subvari- eties of the moduli space SUC(2 ;K ) of rank 2 bundles with canonical deter- minant on a smooth projective curve and 2-divisors, via the inclusion of the moduli space into j2j, singular along the Kummer variety. In particular we show that the moduli space contains all the trisecants of the Kummer and deduce that there are quadrisecant lines only if the curve is hyperelliptic; we show that for generic curves of genus < 6, though no higher, bundles with > 2 sections are cut out by 00; and that for genus 4 this locus is precisely the Donagi-Izadi nodal cubic threefold associated to the curve. This map to projective space (the two cases are of course isomorphic) comes from the complete series on the ample generator of the Picard group, and (at least for a generic curve) is an embedding of the moduli space. Moreover, its image contains much of the geometry studied in connection with the Schottky problem; notably the conguration of Prym-Kummer varieties. In this paper we explore a little of the interplay, via this embedding, between the geometry of vector bundles and the geometry of 2-divisors. On the vector bundle side we are principally concerned with the Brill-Noether loci Wr SU C(2 ;K ) dened by the condition h0(E) >r on stable bundles E. These are analogous to the very classical varieties W r
Kiepert (1873) and Brioschi (1864) published algebraic equations for the n-division points of an elliptic curve, in terms of the Weierstrass ℘-function and its derivatives with respect to a uniformizing
parameter, or another elliptic function, respectively. We generalize both types of formulas for a compact Riemann surface
which, outside from one point, has a smooth polynomial equation in the plane, in the sense that we characterize the points
whose n-th multiple in the Jacobian belongs to the theta divisor.
We set out a method for giving explicit algebraic coordinates on
varieties of elliptic solitons, which consists of finding the
spectral curve by elimination and solving the Jacobi inversion
problem by the use of Kleinian functions and their identities.
As an example we solve a five-particle elliptic Calogero-Moser
system, whose spectral curve turns out to be completely
reducible over three equianharmonic elliptic curves.
A moment map
[(J)\tilde]r :MA ® ([(gl(r))\tilde]+ )*\tilde J_r :M_A \to (\widetilde{gl(r)}^ + )^*
is constructed from the Poisson manifold ([(gl(r))\tilde]+ )*(\widetilde{gl(r)}^ + )^*
of the positive part of the formal loop algebra
[(gl(r))\tilde]\widetilde{gl(r)}
=gl(r)([(gl(r))\tilde]+ )*(\widetilde{gl(r)}^ + )^*
. The pull-back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in ([(gl(r))\tilde]+ )*(\widetilde{gl(r)}^ + )^*
and linearized on the Jacobi variety of an invariant spectral curveX
r
which, generically, is anr-sheeted Riemann surface. Reductions of
[(sl(r,\mathbbC)\tilde])\widetilde{sl(r,\mathbb{C}})
. The theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of PDE's.
We present an operator-coefficient version of Sato's infinite-dimensional
Grassmann manifold, and tau-function. In this context, the Burchnall-Chaundy
ring of commuting differential operators becomes a C*-algebra, to which we
apply the Brown-Douglas-Fillmore theory, and topological invariants of the
spectral ring become readily available. We construct KK classes of the spectral
curve of the ring and, motivated by the fact that all isospectral
Burchnall-Chaundy rings make up the Jacobian of the curve, we compare the
(degree-1) K-homology of the curve with that of its Jacobian. We show how the
Burchnall-Chaundy C*-algebra extension of the compact operators provides a
family of operator-valued tau-functions.
M. Toda in 1967 (\textit{J. Phys. Soc. Japan}, \textbf{22} and \textbf{23})
considered a lattice model with exponential interaction and proved, as
suggested by the Fermi-Pasta-Ulam experiments in the 1950s, that it has exact
periodic and soliton solutions. The Toda lattice, as it came to be known, was
then extensively studied as one of the completely integrable
(differential-difference) non-linear equations which admit exact solutions in
terms of theta functions of hyperelliptic curves. In this paper, we extend
Toda's original approach to give hyperelliptic solutions of the Toda lattice in
terms of hyperelliptic Kleinian (sigma) functions for arbitrary genus. The key
identities are given by generalized addition formulae for the hyperelliptic
sigma functions (J.C. Eilbeck \textit{et al.}, {\it J. reine angew. Math.} {\bf
619}, 2008). We then show that periodic (in the discrete variable, a standard
term in the Toda lattice theory) solutions of the Toda lattice correspond to
the zeros of Kiepert-Brioschi's division polynomials, and note these are
related to solutions of Poncelet's closure problem. One feature of our solution
is that the hyperelliptic curve is related in a non-trivial way to the one
previously used.
In this Letter we obtain a generalization of the Frobenius–Stickelberger addition formula for the (hyperelliptic) s-function of a genus 2 curve in the case of three vector-valued variables. The result is given explicitly in the form of a polynomial in Kleinian P-functions.
In this note, we prove a finiteness result for fibers that are canonical lifts in a given elliptic fibration. The question was motivated by the authors' construction of an arithmetic Euler top, and it highlights an interesting discrepancy between the arithmetic and the classical case: in the former, it is impossible to extend the flows to a compactification of the phase space, viewed as an elliptic fibration over the space of action variables.
The theory of differential equations has an arithmetic analogue in which derivatives of functions are replaced by Fermat quotients of numbers. Many classical differential equations (Riccati, Weierstrass, Painlev\'{e}, etc.) were previously shown to possess arithmetic analogues. The paper introduces an arithmetic analogue of the Euler differential equations for the rigid body.
The zero divisor of the theta function of a compact Riemann surface X of genus g is the canonical theta divisor of Pic up to translation by the Riemann constant for a base point P of X. The complement of the Weierstrass gaps at the base point P given as a numerical semigroup plays an important role, which is called the Weierstrass semigroup. It is classically known that the Riemann constant is a half period for the Jacobi variety of X if and only if the Weierstrass semigroup at P is symmetric. In this article, we analyze the non-symmetric case. Using a semi-canonical divisor , we show a relation between the Riemann constant and a half period of the non-symmetric case. We also identify the semi-canonical divisor for trigonal curves, and remark on an algebraic expression for the Jacobi inversion problem using the relation
The Darboux transformation has been discovered several times in history (cf. references in [EK] and in [G3]): the reason for this is its deep geometric significance. In this note we explore some geometric aspects of the transformation and their relevance to the difficult problems of describing explicitly higher-rank commutative algebras of ordinary differential operators and KP flows.
In this series of three lectures delivered at the NATO Advanced Institute in Vlora, Albania, April 28-May 9, 2008, we present new error-correcting algorithms for geometric Goppa codes, based on rank-2 vector bundles over an algebraic curve. We review the construction of moduli spaces of vector bundles and relevant stratifications and give examples for the Klein curve. We propose questions of algebraic geometry that could be answered using coding theory and vice versa.
In this paper we review some of the recent work on ‘nonabelian theta functions’. We discuss various links between abelian and nonabelian theta functions as well as links with the Schottky problem and open questions.
Some abelian varieties admit several (principal) polarizations. The problem of which Jacobi varieties do, especially among the splittable, challenges transliteration from analysis to algebra. Questions and examples are provided, as well as applications to solution of certain partial differential equations.
To illustrate the use of special functions in physics, we survey the origin and the significance of the theta, sigma, and tau functions in algebraic geometry and partial differential equations. Recent results on the Jacobi inversion problem allow us to construct a new solution to the dispersionless KP equation, defined locally on the Jacobian of any
C
rs
curve. We pose some new questions.
Using Frobenius-Stickelberger-type relations for hyperelliptic curves (Y. Ônishi, Proc. Edinb. Math. Soc. (2) 48 (2005), 705–742), we provide certain addition formulae for any symmetric power of such curves, which hold on the strata Wk, the pre-images in the Jacobian of the classical Wirtinger varieties. In an appendix, we give similar relations for a trigonal curve y
3 = (x – b
1)(x – b
2)(x – b
3)(x – b
4).
Not Available Permanent address: Boston University, Boston, MA 02215, USA.
We consider the linear system |2Θ0| of second order theta functions over the Jacobian JC of a non-hyperelliptic curve C. A result by J. Fay says that a divisor D ∈ |2Θ0| contains the origin script O sign ∈ JC with multiplicity 4 if and only if D contains the surface C - C = {script O sign(p - q) | p, q ∈ C} ⊂ JC. In this paper we generalize Fay's result and some previous work by R.C. Gunning. More precisely, we describe the relationship between divisors containing script O sign with multiplicity 6, divisors containing the fourfold C2 - C2 = {script O sign(p + q - r - s) | p, q, r, s ∈ C}, and divisors singular along C - C, using the third exterior product of the canonical space and the space of quadrics containing the canonical curve. Moreover we show that some of these spaces are equal to the linear span of Brill-Noether loci in the moduli space of semi-stable rank 2 vector bundles with canonical determinant over C, which can be embedded in |2Θ0|.
We study relations among the classical thetanulls of cyclic curves, namely
curves (of genus ) with an automorphism
such that generates a normal subgroup of the group G of
automorphisms, and . Relations between thetanulls
and branch points of the projection are the object of much classical work,
especially for hyperelliptic curves, and of recent work, in the cyclic case. We
determine the curves of genus 2 and 3 in the locus for all G that have a normal subgroup <\s> as above, and all
possible signatures \textbf{C}, via relations among their thetanulls.
In the 1970s, self-similar solutions to integrable PDEs were found to satisfy the Painlevé condition. There ensued a proposal to linearize the Painlevé condition via inverse scattering, which apparently had no sequel. In this paper, we implement an inverse construction for the Baker function and the self-similar condition, to produce by differential algebra the nonlinear ODE through a linear problem.
We investigate geometric properties of the (Sato-Segal-Wilson) Grassmannian and its submanifolds, with special attention to the orbits of the KP flows. We use a coherentstates model, by which Spera and Wurzbacher gave equations for the image of a product of Grassmannians using the Powers-Størmer purification procedure. We extend to this product Sato’s idea of turning equations that define the projective embedding of a homogeneous space into a hierarchy of partial differential equations. We recover the BKP equations from the classical Segre embedding by specializing the equations to finite-dimensional submanifolds.
We revisit the calculation of Calabi’s diastasis function given by Spera and Valli again in the context of C*-algebras, using the τ-function to give an expression of the diastasis on the infinitedimensional Grassmannian; this expression can be applied to the image of the Krichever map to give a proof of Weil’s reciprocity based on the fact that the distance of two points on the Grassmannian is symmetric. Another application is the fact that each (isometric) automorphism of the Grassmannian is induced by a projective transformation in the Plücker embedding.
Verdier's program for classifying elliptic operators with a nontrivial centralizer is outlined. Examples of Boussinesq operators are developed.
We show that commutative rings of formal pseudodifferential operators can be conjugated as subrings in noncommutative Banach algebras of operators in the presence of certain eigenfunctions. Techniques involve those of the Sato Grassmannian as used in the study of the KP hierarchy as well as the geometry of an infinite dimensional Stiefel bundle with structure modeled on such Banach algebras. Generalizations of this procedure are also considered.
The moduli space of curves that admit elliptic subcovers is described in several possible ways: by algebra, group theory,
monodromy, and topology. Some results on the number of components are obtained by combining group theory and topology. Results
surveyed or announced serve to demonstrate the applications of the space under study to integrable systems.
A sequence of geometric Darboux transformations gives a direct link between any KdV Lax operator and any solution of the KP hierarchy whose associated Burchnall-Chaundy ring is ranked 2 and corresponds to a cuspidal or nodal cubic. This gives a geometric explanation for the Bäcklund transformation that relates KdV to the singular Krichever-Novikov (s-KN) equation. Explicit formulas are developed and applied to the example of the Adler-Moser KdV solutions. to produce s-KN and KP solutions of (fake) rank 2. together with their associated Schur partitions.
Comparatively little is known about commutative rings of partial differential operators, while in the ordinary case, concrete
examples and an algebraic(-geometric) structure can be algorithmically determined for large classes. In this note, by the
calculation of the partial μ-shifted differential resultant which we defined in a previous paper, we produce algebraic equations of spectral surfaces
for commutative rings in two variables, and Darboux transformations of Airy-type operators that correspond to morphisms of
surfaces. There are, however, many elementary differential-algebraic statements that we only observe experimentally, thus
we offer open questions which seem to us quite significant in differential algebra, and access to Mathematica code to enable further experimentation.
Motivated by error-correcting coding theory, we pose some hard questions regarding moduli spaces of rank-2 vector bundles over algebraic curves. We propose a new approach to the role of rank-2 bundles in coding theory, using recent results over the complex numbers, namely restriction of vector bundles from the projective space where the curve is embedded. We specialize our analysis to plane quartic curves which, if smooth, are canonical curves of genus three, and remark that all the bundles in question are restrictions. Using the vector-bundle approach, we work out explicit equations for the error divisors viewed as points of a multisecant variety. We specialize canonical quartics even more, to Klein’s curve, and finite fields of characteristic two, a situation in which bundles can be neatly trivialized and codes have been produced. We give explicit equations, work out counting results for curves, Jacobians, and varieties of bundles, revealing several surprising features.