# The Michigan Mathematical Journal

Print ISSN: 0026-2285

Print ISSN: 0026-2285

Publications

Apparently new expressions are given for the exponential of a hermitian matrix,A, in the 2x2,3x3,and 4x4 cases. Replacing A by iA these are explicit formulas for the Fourier transform of exp(iA).

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We work over an algebraically closed ground field of characteristic zero. A
$G$-cover of ${\mathbb P}^1$ ramified at three points allows one to assign to
each finite dimensional representation $V$ of $G$ a vector bundle $\oplus
\mathscr{O}(s_i)$ on ${\mathbb P}^1$ with parabolic structure at the
ramification points. This produces a tensor functor from representation of $G$
to vector bundles with parabolic structure that characterises the original
cover. This work attempts to describe this tensor functor in terms of group
theoretic data. More precisely, we construct a pullback functor on vector
bundles with parabolic structure and describe the parabolic pullback of the
previously described tensor functor.

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We give a method for constructing many examples of automorphisms with positive entropy on rational complex surfaces. The general idea is to begin with a quadratic Cremona transformation that fixes a reduced cubic curve and then use the group structure on the cubic to understand when the indeterminacy and exceptional behavior of the transformation may be eliminated by repeated blowing up. Comment: 29 pages. Many changes, including a new title and the addition of an appendix (contributed by Igor Dolgachev) which carefully treats the group law for singular and reducible cubics. To appear in the Michigan Math Journal

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We construct Abel maps for a stable curve $X$. Namely, for each one-parameter deformation of $X$ with regular total space, and every integer $d>0$, we construct by specialization a map $\alpha^d_X$ from the smooth locus of $X^d$ to the moduli scheme of balanced line bundles on semistable curves over $X$. For $d=1$, we show that $\alpha^1_X$ naturally extends over $X$, and does not depend on the choice of the deformation; we give a precise description of when it is injective.

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We consider the loci of d-elliptic curves in $M_2$, and corresponding loci of
d-elliptic surfaces in $A_2$. We show how a description of these loci as
quotients of a product of modular curves can be used to calculate cohomology of
natural local systems on them, both as mixed Hodge structures and $\ell$-adic
Galois representations. We study in particular the case d=2, and compute the
Euler characteristic of the moduli space of n-pointed bi-elliptic genus 2
curves in the Grothendieck group of Hodge structures.

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We prove a purity theorem for abelian schemes in arbitrary unramified mixed characteristic (0,p). The case p=2 is completely new and the case p>2 fixes several errors in the literature.

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Hodge classes on the moduli space of admissible covers with monodromy group G
are associated to irreducible representations of G. We evaluate all linear
Hodge integrals over moduli spaces of admissible covers with abelian monodromy
in terms of multiplication in an associated wreath group algebra. In case G is
cyclic and the representation is faithful, the evaluation is in terms of double
Hurwitz numbers. In case G is trivial, the formula specializes to the
well-known result of Ekedahl-Lando-Shapiro-Vainshtein for linear Hodge
integrals over the moduli space of curves in terms of single Hurwitz numbers.

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In this paper, we investigate Murre's conjecture on the existence of a Chow--Kuenneth decomposition for a rational homogeneous bundle $Z\to S$ over a smooth variety, defined over complex numbers. Chow-K\"unneth decomposition is exhibited for $Z$ whenever $S$ has a Chow--Kuenneth decomposition. The same conclusion holds for a class of log homogeneous varieties, studied by M. Brion. Comment: Final version, to appear

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Given a smooth non-hyperelliptic prime Fano threefold X, we prove the
existence of all rank 2 ACM vector bundles on X by deformation of semistable
sheaves. We show that these bundles move in generically smooth components of
the corresponding moduli space. We give applications to pfaffian
representations of quartic threefolds in P^4 and cubic hypersurfaces of a
smooth quadric of P^5.

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We consider torus actions on Mori dream spaces and ask whether the associated
Chow quotient is again a Mori dream space and, if so, what does its Cox ring
look like. We provide general tools for the study of these problems and give
solutions for k*-actions on smooth quadrics.

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In this note we prove the following theorem: Let $G$ be a compact Lie group acting on a compact symplectic manifold $M$ in a Hamiltonian fashion. If $L$ is an $l$-dimensional closed invariant submanifold of $M$, on which the $G$-action is locally free then the fundamental class $[L]$ is trivial in $H_l(M,{\mathbb Q})$. We also prove similar results for lower homology groups of $L$, in case the group $G$ is a finite product of copies of $S^1$ and SU(2). The key ingredients of the proofs are Kirwan's theorem that Hamiltonian spaces are equivariantly formal and symplectic reduction.

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We prove the ideal-adic semi-continuity of minimal log discrepancies on
surfaces.

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We characterize ideals whose adjoints are determined by their Rees valuations. We generalize the notion of a regular system of parameters, and prove that for ideals generated by monomials in such elements, the integral closure and adjoints are generated by monomials. We prove that the adjoints of such ideals and of all ideals in twodimensional regular local rings are determined by their Rees valuations. We prove special cases of subadditivity of adjoints. Adjoint ideals and multiplier ideals have recently emerged as a fundamental tool in commutative algebra and algebraic geometry. In characteristic 0 they may be defined using resolution of singularities. In all characteristics, even mixed, Lipman gave the following definition: Definition 0.1: Let R be a regular domain, I an ideal in R. The adjoint adjI of I is defined as follows: adjI = ⋂ {r ∈ R | v(r) ≥ v(I) − v(JRv/R)}, v

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We introduce hyperelliptic simplified (more generally, directed) broken
Lefschetz fibrations, which is a generalization of hyperelliptic Lefschetz
fibrations. We construct involutions on the total spaces of such fibrations of
genus $g\geq 3$ and extend these involutions to the four-manifolds obtained by
blowing up the total spaces. The extended involutions induce double branched
coverings over blown up sphere bundles over the sphere. We also show that the
regular fiber of such a fibration of genus $g\geq 3$ represents a non-trivial
rational homology class of the total space.

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Rapoport and Kottwitz defined the affine Deligne-Lusztig varieties $X_{\tilde{w}}^P(b\sigma)$ of a quasisplit connected reductive group $G$ over $F = \mathbb{F}_q((t))$ for a parahoric subgroup $P$. They asked which pairs $(b, \tilde{w})$ give non-empty varieties, and in these cases what dimensions do these varieties have. This paper answers these questions for $P=I$ an Iwahori subgroup, in the cases $b=1$, $G=SL_2$, $SL_3$, $Sp_4$. This information is used to get a formula for the dimensions of the $X_{\tilde{w}}^K(\sigma)$ (all shown to be non-empty by Rapoport and Kottwitz) for the above $G$ that supports a general conjecture of Rapoport. Here $K$ is a special maximal compact subgroup.

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Work of Kazhdan-Lusztig and Bezrukavnikov suggests the importance of points in affine Springer fibers for which the associated conjugacy class in the finite dimensional Lie algebra is regular. Such points are characterized in a different way in this paper, using the same kind of information as goes into the weight factors used to form weighted orbital integrals.

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We construct explicitly a family of proper subgroups of the tame automorphism
group of affine three-space (in any characteristic) which are generated by the
affine subgroup and a non-affine tame automorphism. One important corollary is
the titular result that settles negatively the open question (in characteristic
zero) of whether the affine subgroup is a maximal subgroup of the tame
automorphism group. We also prove that all groups of this family have the
structure of an amalgamated free product of the affine group and a finite group
over their intersection.

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Given a family of complex affine planes, we show that it is trivial over a Zariski open subset of the base. The proof relies upon a relative version of the contraction theorem. Comment: 12p, LaTeX2e, tar

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We see that a building whose Coxeter group is hyperbolic is itself hyperbolic. Thus any finitely generated group acting co-compactly on such a building is hyperbolic, hence automatic. We turn our attention to affine buildings and consider a group $\Gamma$ which acts simply transitively and in a ``type-rotating'' way on the vertices of a locally finite thick building of type $\tilde A_n$. We show that $\Gamma$ is biautomatic, using a presentation of $\Gamma$ and unique normal form for each element of $\Gamma$, as described in ``Groups acting simply transitively on the vertices of a building of type $\tilde A_n$'' by D.I. Cartwright, to appear, Proceedings of the 1993 Como conference ``Groups of Lie type and their geometries''. Comment: Plain Tex, 12 pages, no figures

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We prove a Corona type theorem with bounds for the Sarason algebra $H^\infty+C$ and determine its spectral characteristics. We also determine the Bass, the dense, and the topological stable ranks of $H^\infty+C$. Comment: v1: 16 pages

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When $G$ is a complex reductive algebraic group and $G/K$ is a reductive symmetric space, the decomposition of $\C[G/K]$ as a $K$-module was obtained (in a non-constructive way) by Richardson, generalizing the celebrated result of Kostant-Rallis for the linearized problem (the harmonic decomposition of the isotropy representation). To obtain a constructive version of Richardson's results, this paper studies the infinite dimensional Lie algebra $\X(G/K)^K$ of $K$-invariant regular algebraic vector fields using the geometry of $G/K$ and the $K$-spherical representations of $G$. Assume $G$ is semisimple and simply-connected and let $\J$ be the algebra of $K$ biinvariant functions on $G$. An explicit set of free generators for the localization $ \X(G/K)^K_{\psi}$ is constructed for a suitable $\psi \in \J$. A commutator formula is obtained for $K$-invariant vector fields in terms of the corresponding $K$-covariant maps from $G$ to the isotropy representation of $G/K$. Vector fields on $G/K$ whose horizontal lifts to $G$ are tangent to the Cartan embedding of $G/K$ into $G$ are called \emph{flat}. When $G$ is simple and simply connected, it is shown that every element of $\X(G/K)^K$ is flat if and only if $K$ is semisimple. The gradients of the fundamental characters of $G$ are shown to generate all conjugation-invariant vector fields on $G$. These results are applied in the case of the adjoint representation of $G = \SL(2,\C)$ to construct a conjugation invariant differential operator whose kernel furnishes a harmonic decomposition of $\C[G]$.

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Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its quadratic closure, we express g_A as a semi-direct product of the well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows us to compute the homotopy Lie algebra associated to the cohomology ring of the complement of a complex hyperplane arrangement, provided some combinatorial assumptions are satisfied. As an application, we give examples of hyperplane arrangements whose complements have the same Poincaré polynomial, the same fundamental group, and the same holonomy Lie algebra, yet different homotopy Lie algebras.

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We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D2n over a global field, which completes some of the main results from the paper “Weakly commensurable arithmetic groups and isospectral locally symmetric spaces ” (Pub. Math. IHES, 2009) by Prasad and Rapinchuk and gives a new proof of a result from another paper by the same authors. One goal of this paper is the (rather technical) Theorem 16 below, which completes some of the main results in the remarkable paper [PrR09] by Gopal Prasad and Andrei Rapinchuk. For example, combining their Theorem 7.5 with our Theorem 16 gives: Theorem 1. Let G1 and G2 be connected absolutely simple algebraic groups over a number field K that have the same K-isomorphism classes of maximal K-tori.

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We examine various versions of oriented cohomology and Borel-Moore homology theories in algebraic geometry and put these two together in the setting of an "oriented duality theory", a generalization of Bloch-Ogus twisted duality theory. This combines and exends work of Panin and Mocanasu. We apply this to give a Borel-Moore homology version $MGL'_{*,*}$ of Voevodsky's $MGL^{*,*}$-theory, and a natural map $\vartheta:\Omega_*\to MGL'_{2*,*}$, where $\Omega_*$ is the algebraic cobordism theory defined by Levine-Morel. We conjecture that $\vartheta$ is an isomorphism and describe a program for proving this conjecture.

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In [DJL07] it was shown that if A is an affine hyperplane arrangement in C^n, then at most one of the L^2-Betti numbers of its complement is non--zero. We will prove an analogous statement for complements of any algebraic curve in C^2. Furthermore we also recast and extend results of [LM06] in terms of L^2-Betti numbers.

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In this note we study two features of submanifolds (subvarieties) with ample
normal bundles in a compact K\"ahler manifold X. First, we study how algebraic
X can be, i.e. we investigate the algebraic dimension. Second, we study curves
with ample normal bundles in case X is projective. We prove in various cases
that the class of the curve is in the interior of the Mori cone.

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We investigate the general structure of the automorphism group and the Lie algebra of derivations of a finitely generated vertex operator algebra. The automorphism group is isomorphic to an algebraic group. Under natural assumptions, the derivation algebra has an invariant bilinear form and the ideal of inner derivations is nonsingular.

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We study initial algebras of determinantal rings, defined by minors of
generic matrices, with respect to their classical generic point. This approach
leads to very short proofs for the structural properties of determinantal
rings. Moreover, it allows us to classify their Cohen-Macaulay and Ulrich
ideals.

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For homomorphism K-->S of commutative rings, where K is Gorenstein and S is essentially of finite type and flat as a K-module, the property that all non-trivial fiber rings of K-->S are Gorenstein is characterized in terms of properties of the cohomology modules Ext_n^{S\otimes_KS}S{S\otimes_KS}.

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We say that a Hopf algebra has the Chevalley property if the tensor product of any two simple modules over this Hopf algebra is semisimple. In this paper we classify finite dimensional triangular Hopf algebras with the Chevalley property, over the field of complex numbers. Namely, we show that all of them are twists of triangular Hopf algebras with R-matrix having rank <=2, and explain that the latter ones are obtained from group algebras of finite supergroups by a simple modification procedure. We note that all examples of finite dimensional triangular Hopf algebras which are known to the authors, do have the Chevalley property, so one might expect that our classification potentially covers all finite dimensional triangular Hopf algebras.

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We study the approximation of J-holomorphic maps continuous to the boundary from ma domain in the complex plane into an almost complex manifold by maps J-holomorphic to the boundary, giving partial results in the non-integrable case. For the integrable case, we study arcs in complex manifolds and establish the existence of neighborhoods biholomorphic to open sets in Euclidean space for several classes of arcs. As an application, we obtain $\mathcal{C}^k$ approximation of holomorphic maps continuous to the boundary into complex manifolds by maps holomorphic to the boundary, provided the boundary is nice enough.

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We prove that a relatively compact pseudoconvex domain with smooth boundary in an almost complex manifold admits a bounded strictly plurisubharmonic exhaustion function. We use this result for the study of convexity and hyperbolicity properties of these domains and the contact geometry of their boundaries.

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If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct product SL(2,R)^m x SL(2,C)^n$, with m + n > 1. (In geometric terms, this can be interpreted as a statement about the existence of totally geodesic subspaces of finite-volume, noncompact, locally symmetric spaces of higher rank.) Another formulation of the result states that if G is any isotropic, almost simple algebraic group over Q (the rational numbers), such that the real rank of G is greater than 1, then G contains an isotropic, almost simple Q-subgroup H, such that H is quasisplit, and the real rank of H is greater than 1.

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The length is(w) of the longest increasing subsequence of a permutation w in the symmetric group S_n has been the object of much investigation. We develop comparable results for the length as(w) of the longest alternating subsequence of w, where a sequence a,b,c,d,... is alternating if a>b d<.... For instance, the expected value (mean) of as(w) for w in S_n is exactly (4n+1)/6 if n>1.

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Monge-Ampere currents generated by plurisubharmonic functions of logarithmic growth are studied. Upper bounds for their total masses are obtained in terms of growth characteristics of the functions. In particular, this gives a plurisubharmonic version of D.Bernstein's theorem on the number of zeros of polynomial mappings in terms of the Newton polyhedra, and a representation for Demailly's generalized degrees of plurisubharmonic functions.

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Given an algebraic torus action on a normal projective variety with finitely generated total coordinate ring, we study the GIT-equivalence for not necessarily ample linearized divisors, and we provide a combinatorial description of the partially ordered set of GIT-equivalence classes. As an application, we extend in the $\QQ$-factorial case a basic feature of the collection of ample GIT-classes to the partially ordered collection of maximal subsets with a quasiprojective quotient: for any two members there is at most one minimal member comprising both of them. Moreover, we demonstrate in an example, how our theory can be applied for a systematic treatment of ``exotic projective orbit spaces'', i.e., projective geometric quotients that do not arise from any linearized ample divisor.

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Let M be a real analytic strictly pseudoconvex manifold of higher codimension in complex space, and let M' be the cartesian product of two or more compact real analytic strictly convex hypersurfaces. We prove that a germ of a biholomorphic map taking an open set in M to M' extends as a biholomorphic map along any path in M.

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We study biorthogonal sequences with special properties, such as weak or weak-star convergence to 0, and obtain an extension of the Josefson-Nissenzweig theorem. This result is applied to embed analytic disks in the fiber over 0 of the spectrum of H^infinity (B), the algebra of bounded analytic functions on the unit ball B of an arbitrary infinite dimensional Banach space. Various other embedding theorems are obtained. For instance, if the Banach space is superreflexive, then the unit ball of a Hilbert space of uncountable dimension can be embedded analytically in the fiber over 0 via an embedding which is uniformly bicontinuous with respect to the Gleason metric.

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We prove and organize some results on the normal forms of Hermitian operators
composed with the Veronese map. We apply this general framework to prove two
specific theorems in CR geometry. First, extending a theorem of Faran, we
classify all real-analytic CR maps between any hyperquadric in $\C^2$ and any
hyperquadric in $\C^3$, resulting in a finite list of equivalence classes.
Second, we prove that all degree-two CR maps of spheres in all dimensions are
spherically equivalent to a monomial map, thus obtaining an elegant
classification of all degree-two CR sphere maps.

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We prove Angehrn-Siu type effective base point freeness and point separation for log canonical pairs. Comment: 12 pages, v2: minor modifications, v3: very minor modifications, v4: title changed, v5: revision following referee's comments, v6: very minor modifications, to appear in Michigan Math. J

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We give a new construction of slice knots via annulus twists which provides
potential counterexamples to the slice-ribbon conjecture. Among them, the knots
constructed by Omae are the simplest ones. In our previous work, we proved that
one of them is a ribbon knot. In this paper, we prove that the rest are also
ribbon knots.

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If S is a hyperbolic surface and S̊. The surface obtained from S by removing a point. The mapping class groups Mod(S) and Mod(S̊) fit into a short exact sequence 1→π1(S)→Mod(S̊)→Mod(S) →1 We give a new criterion formapping classes i. The kernel to be pseudo- Anosov usin. The geometry of hyperbolic 3-manifolds. Namely, we show that if M is an ε-thick hyperbolic manifold homeomorphic to S × R, then an element of π1(M) ≅ π1(S) represents a pseudo- Anosov element of Mod( . S) if its geodesic representative is "wide". We establish similar criteria where M is replaced with a coarsely hyperbolic surface bundle coming from a δ-hyperbolic surface-group extension.

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A polynomial is a direct sum if it can be written as a sum of two non-zero
polynomials in some distinct sets of variables, up to a linear change of
vriables. We analyse criteria for a homogeneous polynomial to be decomposable
as a direct sum, in terms of the apolar ideal of the polynomial. We prove that
the apolar ideal of a polynomial of degree $d$ strictly depending on all
variables has a minimal generator of degree $d$ if and only if it is a limit of
direct sums.

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I give various criteria for singularities to appear on geometric generic fibers of morphism between smooth schemes in positive characteristics. This involves local fundamental groups, jacobian ideals, projective dimension, tangent and cotangent sheaves, and the effect of Frobenius. As an application, I determine which rational double points do appear on geometric generic fibers.

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Under natural hypotheses we give an upper bound on the dimension of families of singular curves with hyperelliptic normalizations on a surface S with p_g(S) >0 via the study of the associated families of rational curves in Hilb^2(S). We use this result to prove the existence of nodal curves of geometric genus 3 with hyperelliptic normalizations, on a general K3 surface, thus obtaining specific 2-dimensional families of rational curves in its Hilbert square. We describe two infinite series of examples of general, primitively polarized K3's such that their Hilbert squares contain a IP^2 or a threefold birational to a IP^1-bundle over a K3. We discuss some consequences on the Mori cone of the Hilbert square of a general K3.

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We prove formulas for the core of ideals that apply in arbitrary characteristic.

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We provide a sufficient condition for a full subcomplex of the arc complex
for a compact orientable surface to be contractible, which generalizes the
result by Hatcher that the arc complexes are contractible. As an application,
we construct infinitely many Heegaard splittings whose sphere complexes are
contractible, including the genus-$2$ Heegaard splitting of $S^2 \times S^1$.
Further, if a Heegaard splitting is obtained by gluing a splitting of Hempel
distance at least $4$ and the genus-$1$ splitting of $S^2 \times S^1$, we show
that the Goeritz group of the splitting is finitely generated.

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We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and hence give rise to weight systems. Among these weight systems are those associated with the Conway and HOMFLYPT polynomials. We extend these ideas to looking at a space of {\it marked} chord diagrams modulo an extended set of 2-term relations, define a set of generators for this space, and again derive weight systems from the adjacency matrices of the (marked) intersection graphs. Among these weight systems are those associated with the Kauffman polynomial.

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Let f : X --> X be a dominant rational map of a projective variety defined
over a number field. An important geometric-dynamical invariant of f is its
(first) dynamical degree d_f= lim SpecRadius((f^n)^*)^{1/n}. For algebraic
points P of X whose forward orbits are well-defined, there is an analogous
(upper) arithmetic degree a_f(P) = limsup h_X(f^n(P))^{1/n}, where h_X is an
ample Weil height on X. In an earlier paper, we proved the fundamental
inequality a_f(P) \le d_f and conjectured that a_f(P) = d_f whenever the orbit
of P is Zariski dense. In this paper we show that the conjecture is true for
several types of maps. In other cases, we provide support for the conjecture by
proving that there is a Zariski dense set of points with disjoint orbits and
satisfying a_f(P) = d_f.

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This paper is concerned with the arithmetic of the elliptic K3 surface with configuration [1,1,1,12,3*]. We determine the newforms and zeta-functions associated to X and its twists. We verify conjectures of Tate and Shioda for the reductions of X at 2 and 3.

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