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We consider birational projective contractions f:X -> Y from a smooth symplectic variety X over the complex numbers. We first show that exceptional rational curves on X deform in a family of dimension at least 2n-2. Then we show that these contractions are generically coisotropic, provided X is projective. Then we specialize to contractions with 1-dimensional exceptional fibres. We classify them in a natural way in terms of (\Gamma, G), where \Gamma is a Dynkin diagram of type A_l, D_l or E_l and G is a permutation group of automorphisms of \Gamma. The 1-dimensional fibres do not degenerate, except if the contraction is of type (A_{2l},S_2). In that case they do not degenerate in codimension 1. Furthermore we show that the normalization of any irreducible component of Sing(Y) is a symplectic variety. We also provide examples for contractions of any type (\Gamma, G). | Contractions of Symplectic Varieties | 12,100 |
A generic quartic 3-fold X admits a 7-dimensional family of representations as the Pfaffian of an 8 by 8 skew-symmetric matrix of linear forms. This provides a 7-dimensional moduli space M of rank 2 vector bundles on X. A precise geometric description of a 14-dimensional family of half-canonical curves C of genus 15 in X such that the above vector bundles are obtained by Serre's construction from C is given. It is proved that the Abel-Jacobi map of this family factors through M, and the resulting map from M to the intermediate Jacobian is quasi-finite. In particular, every component of M has non-negative Kodaira dimension. Some other constructions of rank 2 vector bundles with small Chern classes are discussed; it is proved that the smallest possible charge of an instanton on X is 4. | Quartic 3-fold: Pfaffians, instantons and half-canonical curves | 12,101 |
We define a certain compactifiction of the general linear group and give a modular description for its points with values in arbitrary schemes. This is a first step in the construction of a higher rank generalization of Gieseker's degeneration of moduli spaces of vector bundles over a curve. We show that our compactification has similar properties as the ``wonderful compactification'' of algebraic groups of adjoint type as studied by de Concini and Procesi. As a byproduct we obtain a modular description of the points of the wonderful compactification of $\text{PGl}_n$. | A Modular Compactification of the General Linear Group | 12,102 |
In this paper we show a general method to compactify certain open varieties by adding normal crossing divisors. This is done by proving that {\it blowing up along an arrangement of subvarieties} can be carried out. Important examples such as Ulyanov's configuration spaces, spaces of holomorphic maps, etc., are covered. Intersection ring and (non-recursive) Hodge polynomails are computed. Further general structures arising from the blowup process are described and studied. | A Compactification of Open Varieties | 12,103 |
Let \alpha be a Schur root; let h=hcf_v(\alpha(v)) and let p = 1 - < \alpha/h,\alpha/h >. Then a moduli space of representations of dimension vector \alpha is birational to p h by h matrices up to simultaneous conjugacy. Therefore, if h=1,2,3 or 4, then such a moduli space is a rational variety and if h divides 420 it is a stably rational variety. | Birational classification of moduli spaces of representations of quivers | 12,104 |
The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blow-ups and blow-downs, and get an easy criterion to determine whether a given nonsingular toric variety is a Fano variety or not. As applications of these results, we get a toric version of a theorem of Mori, and can classify, in principle, all nonsingular toric Fano varieties obtained from a given nonsingular toric Fano variety by finite successions of equivariant blow-ups and blow-downs through nonsingular toric Fano varieties. Especially, we get a new method for the classification of nonsingular toric Fano varieties of dimension at most four. These methods are extended to the case of Gorenstein toric Fano varieties endowed with natural resolutions of singularities. Especially, we easily get a new method for the classification of Gorenstein toric Fano surfaces. | Toward the classification of higher-dimensional toric Fano varieties | 12,105 |
Let X be a projective geometrically irreducible non-singular algebraic curve defined over a finite field F of order $q^2$. If the number of F-rational points of X satisfies the Hasse-Weil upper bound, then X is said to be F-maximal. For a point P_0\in X(F), let \pi be the morphism arising from the linear series D:=|(q+1)P_0|, and let N:=dim(D). It is known that N\ge 2 and that \pi is independent of P_0 whenever X is F-maximal. The following theorems will be proved: Theorem 0.1: If X is F-maximal, then \pi:X\to \pi(X) is a F-isomorphism. The non-singular model \pi(X) has degree q+1 and lies on a Hermitian variety defined over F of P^N(\bar F); Theorem 0.2: If X is F-maximal, then it is F-isomorphic to a curve Y in P^M(\bar F), with 2\le M\le N, such that Y has degree q+1 and lies on a non-degenerate Hermitian variety defined over F of \P^M(\bar F). Furthermore, Aut_F(X) is isomorphic to a subgroup of the projective unitary group PGU(M+1,q^2); Theorem 0.3: If X is F-birational to a curve Y embedded in P^M(\bar F) such that Y has degree q+1 and lies on a non-degenerate Hermitian variety defined over F of P^M(\bar F), then X is F-maximal and X is F-isomorphic to Y. | Embedding of a maximal curve in a Hermitian variety | 12,106 |
We sharpen the two main tools used to treat the compactified Jacobian of a singular curve: Abel maps and presentation schemes. First we prove a smoothness theorem for bigraded Abel maps. Second we study the two complementary filtrations provided by the images of certain Abel maps and certain presentation schemes. Third we study a lifting of the Abel map of bidegree (m,1) to the corresponding presentation scheme. Fourth we prove that, if a curve is blown up at a double point, then the corresponding presentation scheme is a IP^1-bundle. Finally, using Abel maps of bidegree (m,1), we characterize the curves having double points at worst | Abel Maps and Presentation Schemes | 12,107 |
We prove the following autoduality theorem for an integral projective curve C in any characteristic. Given an invertible sheaf L of degree 1, form the corresponding Abel map A_L: C->J, which maps C into its compactified Jacobian, and form its pullback map A_L^*: Pic^0_J to J, which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, points of multiplicity 2, then A_L^* is an isomorphism, and forming it commutes with specializing C. Much of our work is valid, more generally, for a family of curves with, at worst, points of embedding dimension 2. In this case, we use the determinant of cohomology to construct a right inverse to A_L^*. Then we prove a scheme-theoretic version of the theorem of the cube, generalizing Mumford's, and use it to prove that A_L^* is independent of the choice of L. Finally, we prove our autoduality theorem: we use the presentation scheme to achieve an induction on the difference between the arithmetic and geometric genera; here, we use a few special properties of points of multiplicity 2. | Autoduality of the compactified Jacobian | 12,108 |
We show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations. These were first constructed as flips in the case of curves by M. Thaddeus via Geometric Invariant Theory, and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper. We expose the finer structure of a second flip; again for varieties of arbitrary dimension. We also prove a result on the cubic generation of the secant variety and give some conjectures on the behavior of equations defining the higher secant varieties. | Secant Varieties and Birational Geometry | 12,109 |
We study the parametrization of the moduli space Bun_2(C)_L of rank 2 bundles over a curve C with fixed determinant, provided by Hecke modifications at fixed points of the trivial bundle. This parametrization is closely related to the Tyurin parametrization of vector bundles over curves. We use it to parametrize the Hitchin and KZB systems, as well as lifts of the Beilinson-Drinfeld D-modules. We express a generating series for the lifts of the Beilinson-Drinfeld operators in terms of a "quantum L-operator" \ell(z). We explain the relation to earlier joint work with G. Felder, based on parametrization by flags of bundles (math/9807145) and introduce filtrations on conformal blocks, related with the Hecke modifications. | Hecke-Tyurin parametrization of the Hitchin and KZB systems | 12,110 |
In 70's there was discovered a construction how to attach to some algebraic-geometric data an infinite-dimensional subspace in the space k((z)) of the Laurent power series. The construction was successfully used in the theory of integrable systems, particularly, for the KP and KdV equations. There were also found some applications to the moduli of algebraic curves. But there remained a hard restriction by the case of curves, so by dimension 1. Recently, it was pointed out by the author that there are some connections between the theory of the KP-equations and the theory of n-dimensional local fields. From this point of view it becomes clear that the Krichever construction should have a generalization to the case of higher dimensions. This generalization is suggested in the paper for the case of algebraic surfaces. | Krichever Correspondence for Algebraic Surfaces | 12,111 |
Vladimir Shpilrain and Jie-Tai Yu have asked for an effective algorithm to decide if two elements of C[x,y] are related by an automorphism of C[x,y]. We describe here an efficient algorithm that decides this question and finds the automorphism if it exists. The algorithm is due to the second author, who described it in terms of Newton polygons, with C replaceable by a field of any characteristic. Here we describe it in terms of splice diagrams, which gains some efficiency at cost of generality (it currently applies only to characteristic zero). Part of the purpose of this paper is to give an exposition of the use of splice diagrams in studying C[x,y]. | Algorithms for polynomials in two variables | 12,112 |
We show the density of the jumping loci of the Picard number of a hyperk\"ahler manifold under small deformation and provide several applications. In particular, we apply this to reveal the structure of hierarchy among all the narrow Mordell-Weil lattices of Jacobian K3 surfaces. | Families of hyperkähler manifolds | 12,113 |
We characterize contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. Using this we generalize the classical criteria of Castelnuovo and Artin. As application we derive a finiteness result on homogeneous spectra defined by Weil divisors on surfaces. | On contractible curves on normal surfaces | 12,114 |
We prove some lower bounds on certain twists of the canonical bundle of a codimension-2 subvariety of a generic hypersurface in projective space. In particular we prove that the generic sextic threefold contains no rational or elliptic curves and no nondegenerate curves of genus 2. | Beyond a conjecture of Clemens | 12,115 |
We classify singular fibres over general points of the discriminant locus of projective complex Lagrangian fibrations on 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type A_n, $\tilde{A_n}$, $\tilde{D_n}$ or D_n. | On singular fibres of complex Lagrangian fibrations | 12,116 |
Let M be a quasiprojective algebraic manifold with K_M=0 and G a finite automorphism group of M acting trivially on the canonical class K_M; for example, a subgroup G of SL(n,C) acting on C^n in the obvious way. We aim to study the quotient variety X=M/G and its resolutions Y -> X (especially under the assumption that Y has K_Y=0) in terms of G-equivariant geometry of M. At present we know 4 or 5 quite different methods of doing this, taken from string theory, algebraic geometry, motives, moduli, derived categories, etc. For G in SL(n,C) with n=2 or 3, we obtain several methods of cobbling together a basis of the homology of Y consisting of algebraic cycles in one-to-one correspondence with the conjugacy classes or the irreducible representations of G. | La correspondance de McKay | 12,117 |
By associating a `motivic integral' to every complex projective variety X with at worst canonical, Gorenstein singularities, Kontsevich proved that, when there exists a crepant resolution of singularities Y of X, the Hodge numbers of Y do not depend upon the choice of the resolution. In this article we provide an elementary introduction to the theory of motivic integration, leading to a proof of the result described above. We calculate the motivic integral of several quotient singularities and discuss these calculations in the context of the cohomological McKay correspondence. | An introduction to motivic integration | 12,118 |
Let X be a quasiprojective smooth surface defined over an algebraically closed field of positive characteristic. We show that if X is Frobenius split then so is the Hilbert scheme Hilb^n(X) of n points in X. In particular, we get the higher cohomology vanishing for ample line bundles on Hilb^n(X) when X is projective and Frobenius split. | Frobenius splitting of Hilbert schemes of points on surfaces | 12,119 |
Locally analytically, any isolated double point occurs as a double covering of a smooth surface. It can be desingularized via the canonical resolution, as it is well-known. In this paper we explicitly compute the fundamental cycle of both the canonical and minimal resolution of a double point singularity and we classify those for which the fundamental cycle differs from the fiber cycle. Finally we compute the conditions that a double point imposes to pluricanonical systems. | Explicit Resolutions of Double Point Singularities of Surfaces | 12,120 |
Let $\sE$ be an ample rank $r$ bundle on a smooth toric projective surface, $S$, whose topological Euler characteristic is $e(S)$. In this article, we prove a number of surprisingly strong lower bounds for $c_1(\sE)^2$ and $c_2(\sE)$. We also enumerate the exceptions to either the inequality $c_1(\sE)^2\ge 4e(S)$ or the inequality $c_2(\sE)\ge e(S)$ holding. | Chern Numbers of Ample Vector Bundles on Toric Surfaces | 12,121 |
Roughly speaking, a conic bundle is a surface, fibered over a curve, such that the fibers are conics (not necessarily smooth). We define stability for conic bundles and construct a moduli space. We prove that (after fixing some invariants) these moduli spaces are irreducible (under some conditions). Conic bundles can be thought of as generalizations of orthogonal bundles on curves. We show that in this particular case our definition of stability agrees with the definition of stability for orthogonal bundles. Finally, in an appendix by I. Mundet i Riera, a Hitchin-Kobayashi correspondence is stated for conic bundles. | Stability of conic bundles (with an appendix by Mundet i Riera) | 12,122 |
This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector budles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory. | Algebraic stacks | 12,123 |
We discuss the rigidity problem for Mori fibrations on del Pezzo surfaces of degree 1, 2 and 3 over ${\mathbb P}^1$ and formulate the following conjecture: such a del Pezzo fibration $V/{\mathbb P}^1$ is birationally rigid if and only if its quasi-effective and adjunction thresholds coincide. We prove the "only if" part of this conjecture. | On a rigidity criterion for del Pezzo fibrations over ${\mathbb P}^1$ | 12,124 |
In this paper we prove two results concerning the classification of Siegel modular threefolds. Let A_{1,d}(n) be the moduli space of abelian surfaces with a (1,d)-polarization and a full level-n structure and let A_{1,d}^{lev}(n) be the space where one has fixed an additional canonical level structure. We prove that A_{1,d}(n) is of general type if (d,n)=1 and n ist at least 4. This is the best possible result which one can prove for all d simultaneously. Let p be an odd prime and assume that (p,n)=1. Then we prove that the Voronoi compactification of A_{1,p}^{lev}(n) is smooth and has ample canonical bundle if and only if n is greater than or equal to 5. | Igusa's modular form and the classification of Siegel modular threefolds | 12,125 |
Let X be as smooth complex projective variety with Neron-Severi group isomorphic to Z, and D an irreducible divisor with normal crossing singularities. Assume r is equal to 2 or 3. We prove that if the fundamental group of X doesn't have irreducible PU(r) representations, then the fundamental group of X-D doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth and X is a complex surface. | Parabolic bundles and representations of the fundamental group | 12,126 |
Let $f: X \to S$ be a smooth morphism in characteristic 0, and let $(E, \nabla_{X/S})$ be a relative regular connection. We define a cohomology of relative differential characters on $X$ which receives classes of $(E, \nabla_{X/S})$. It says in particular that the partial vanishing of the trace of the iterated Atiyah classs can be made canonical. While applied to a family of curves and $c_2$, the construction yields a connection of $f_*c_2(E)$. This one has been constructed analytically by A. Beilinson. We also relate our construction to the trace complex of Beilinson of Schechtman. | Relative Algebraic Differential Characters | 12,127 |
We construct all codimension 1 multi-germs of maps (k^n,T)-->(k^p,0) with n > p-2, (n,p) nice dimensions, k = R or C, by augmentation and concetenation operations, starting from mon-germs (|T|=1). As an application, we prove general results for multi-germs of corank <2: every one has a real form with real perturbation carrying the vanishing homology of the complexification, every one is quasihomogeneous, and when n=p-1 every one has image Milnor number equal to 1 (the comparable result for n>p-1 being already known). | Vanishing topology of codimension 1 multi-germs over R and C | 12,128 |
Let $\mathcal N$ be the moduli space of sextics with 3 (3,4)-cusps. The quotient moduli space ${\mathcal N}/G$ is one-dimensional and consists of two components, ${\mathcal N}_{torus}/G$ and ${\mathcal N}_{gen}/G$. By quadratic transformations, they are transformed into one-parameter families $C_s$ and $D_s$ of cubic curves respectively. We study the Mordell-Weil torsion groups of cubic curves $C_s$ over $\bfQ$ and $D_s$ over $\bfQ(\sqrt{-3})$ respectively. We show that $C_{s}$ has the torsion group $\bf Z/3\bf Z$ for a generic $s\in \bf Q$ and it also contains subfamilies which coincide with the universal families given by Kubert with the torsion groups $\bf Z/6\bf Z$, $\bf Z/6\bf Z+\bf Z/2\bf Z$, $\bf Z/9\bf Z$ or $\bf Z/12\bf Z$. The cubic curves $D_s$ has torsion $\bf Z/3\bf Z+\bf Z/3\bf Z$ generically but also $\bf Z/3\bf Z+\bf Z/6\bf Z$ for a subfamily which is parametrized by $ \bf Q(\sqrt{-3}) $. | Elliptic Curves from Sextics | 12,129 |
Unitary representations of the fundamental group of a Kahler manifold correspond to polystable vector bundles (with vanishing Chern classes). Semisimple linear representations correspond to polystable Higgs bundles. In this paper we find the objects corresponding to affine representations: the linear part gives a Higgs bundle and the translation part corresponds to an element of a generalized de Rham cohomology. | Affine representations of the fundamental group (with an appendix on
parabolic representations) | 12,130 |
Borrowing a reduction principle to a recent preprint of G. Faltings (toroidal resolution of some matrix singularities, 1999), we use Lafforgue's compactification of PGL_r^{N+1}/PGL_r to construct a canonical log-smooth toroidal resolution for the bad reduction in a prime p of Shimura varieties of unitary and symplectic type with parahoric level structures at p. Using this result, non-canonical semi-stable resolutions over Z_p[p^{1/\nu}] can be derived. | A toroidal resolution for the bad reduction of some Shimura varieties | 12,131 |
Suppose that X' is a smooth affine algebraic variety of dimension 3 with H_3(X')=0 which is a UFD and whose invertible functions are constants. Suppose that Z is a Zariski open subset of X which has a morphism p : Z -> U into a curve U such that all fibers of p are isomorphic to C^2. We prove that X' is isomorphic to C^3 iff none of irreducible components of X'-Z has non-isolated singularities. Furthermore, if X' is C^3 then p extends to a polynomial on C^3 which is linear in a suitable coordinate system. As a consequence we obtain the fact formulated in the title of the paper. | Polynomials with general C^2-fibers are variables. I | 12,132 |
Consider a PEL-Shimura variety associated to a unitary group that splits over an unramified extension of Q_p. Rapoport and Zink have defined a model of the Shimura variety over the ring of integers of the completion of the reflex field at a place lying over p, with parahoric level structures at p. We show that this model is flat, as conjectured by Rapoport and Zink, and that its special fibre is reduced. | On the flatness of models of certain Shimura varieties of PEL-type | 12,133 |
For any odd $n$, we describe a smooth minimal (i.e. obtained by adding an irreducible hypersurface) compactification $\tilde S_n$ of the quasi-projective homogeneous variety $S_{n}=PGL(n+1)/SL(2)$ that parameterizes the rational normal curves in $P^n$. We show that $\tilde S_{n}$ is isomorphic to a component of the Maruyama scheme of the semi-stable sheaves on $P^n$ of rank $n$ and Chern polynomial $(1+t)^{n+2}$ and we compute its Betti numbers. In particular $\tilde S_{3}$ is isomorphic to the variety of nets of quadrics defining twisted cubics, studied by G. Ellinsgrud, R. Piene and S. Str{\o}mme (Space curves, Proc. Conf., LNM 1266). | On a compactification of the moduli space of the rational normal curves | 12,134 |
Let $G$ be a reductive group, let $Gr=G((t))/G[[t]]$ be the corresponding affine Grassmannian and let $Fl=G((t))/I$ be the affine flag variety. We construct, following an idea of Belinson, a 1-parametric deformation of the product $Gr\times G/B$ to $Fl$. We use this construction to produce perverse sheaves on $Fl$ which are central with respect to the convolution product from spherical perverse sheaves on $Gr$. | Construction of central elements in the affine Hecke algebra via nearby
cycles | 12,135 |
This paper has been superseded to a great extent by the following: Paper: math.AG/0511558 Title: The Neron-Severi group of a proper seminormal complex variety Authors: L. Barbieri-Viale, A. Rosenschon & V. Srinivas Paper: math.AG/0102150 Title: Deligne's Conjecture on 1-Motives Authors: Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito Journal-ref: Ann. of Math. (2), Vol. 158 (2003), no. 2, 593--633 | On algebraic mixed Hodge substructures of H^2 | 12,136 |
The `linear orbit' of a plane curve of degree $d$ is its orbit in $\P^{d(d+3)/2}$ under the natural action of $\PGL(3)$. In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane curve, and give explicit formulas for plane curves with irreducible singularities. The main tool is an intersection@-theoretic study of the projective normal cone of a scheme determined by the curve in the projective space $\P^8$ of $3\times 3$ matrices; this expresses the degree of the orbit closure in terms of the degrees of suitable loci related to the limits of the curve. These limits, and the degrees of the corresponding loci, have been established in previous work. | Linear orbits of arbitrary plane curves | 12,137 |
Gau{\ss}-Manin determinant connections associated to irregular connections on a curve are studied. The determinant of the Fourier transform of an irregular connection is calculated. The determinant of cohomology of the standard rank 2 Kloosterman sheaf is computed modulo 2 torsion. Periods associated to irregular connections are studied in the very basic $\exp(f)$ case, and analogies with the Gau{\ss}-Manin determinant are discussed. | Gauß-Manin determinant connections and periods for irregular
connections | 12,138 |
We introduce a variant of the usual Kahler forms on free and almost free divisors and their deformations, and show that they enjoy the same depth properties as Kahler forms on isolated complete intersection singularities. Using these forms, it is possible to describe analytically the vanishing cohomology in families of free divisors, in precise analogy with the classical description for the Milnor fibration of an ICIS, due to Brieskorn and Greuel. This applies in par- ticular to the family of discriminants of a versal deformation of an unstable map-germ. | Differential forms on free and almost free divisors | 12,139 |
We study non-isotrivial projective families of elliptic surfaces of Kodaira dimension one, over complex projective curves. If the base is an elliptic curve, we show that the family must have a singular fibre, and that over the projective line it must have at least three singular fibres. Similar results, for families of surfaces of general type, have been obtained by Migliorini and Kov\'acs, and they are well-known for projective families of surfaces of Kodaira dimension zero. Revised version: We corrected some minor errors and ambiguities, and we completed the list of references. | On the isotriviality of families of elliptic surfaces | 12,140 |
We prove canonical isomorphisms between Spin Verlinde spaces, i.e, spaces of global sections of a determinant line bundle over the moduli space of semistable Spin-bundles over a smooth projective curve C, and the dual spaces of theta functions over Prym varieties of unramified double covers of C. | A duality for Spin Verlinde spaces and Prym theta functions | 12,141 |
We study aspects related to Kontsevich's homological mirror symmetry conjecture in the case of Calabi-Yau complete intersections in toric varieties. In a 1996 lecture at Rutgers University, Kontsevich indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in the homological mirror symmetry conjecture should also be identified. Our main results provide an explicit geometric construction of the correspondence between the automorphisms of the two types of categories. | Hypergeometric functions and mirror symmetry in toric varieties | 12,142 |
We introduce symmetrizing operators of the polynomial ring $A[x]$ in the varible $x$ over a ring $A$. When $A$ is an algebra over a field $k$ these operators are used to characterize the monic polynomials $F(x)$ of degree $n$ in $A[x]$ such that $A\otimes_k k[x]_{(x)}/(F(x))$ is a free $A$-module of rank $n$. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length $n$ of $k[x]_{(x)}$. | The Hilbert scheme parameterizing finite length subschemes of the line
with support at the origin | 12,143 |
Let $k[x]_{(x)}$ be the polynomial ring $k[x]$ localized in the maximal ideal $(x)\subseteq k[x]$. We study the Hilbert functor parameterizing ideals of colength $n$ in this ring {\it having support at the origin}. The main result of this article is that this functor is not representable. We also give a complete description of the functor as a limit of representable functors. | On the representability of ${\Cal Hilb}^nk[x]_{(x)}$ | 12,144 |
Let A[X]_U be a fraction ring of the polynomial ring A[X] in the variable X over a commutative ring A. We show that the Hilbert functor {Hilb}^n_{A[X]_U} is represented by an affine scheme $\text{Symm}^n_A(A[X]_U)$ give as the ring of symmetric tensors of $\otimes_A^nA[X]_U$. The universal family is given as $\text{Symm}^{n-1}_A(A[X]_U)\times_A \text{Spec}(A[X]_U)$. | Symmetric tensors with applications to Hilbert schemes | 12,145 |
The "peak reduction" method is a powerful combinatorial technique with applications in many different areas of mathematics as well as theoretical computer science. It was introduced by Whitehead, a famous topologist and group theorist, who used it to solve an important algorithmic problem concerning automorphisms of a free group. Since then, this method was used to solve numerous problems in group theory, topology, combinatorics, and probably in some other areas as well. In this paper, we give a survey of what seems to be the first applications of the peak reduction technique in commutative algebra and affine algebraic geometry. | Peak reduction technique in commutative algebra | 12,146 |
We construct inductively an equivariant compactification of the algebraic group ${\mathbb W}_n$ of Witt vectors of finite length over a field of characteristic $p>0$. We obtain smooth projective rational varieties $\bar{\mathbb W}_n$, defined over $\mathbf F_p$; the boundary is a divisor whose reduced subscheme has normal crossings. The Artin-Schreier-Witt isogeny $F-1:{\mathbb W}_n\to {\mathbb W}_n$ extends to a finite cyclic cover ${\mathbf\Psi}_n:\bar{\mathbb W}_n\to \bar{\mathbb W}_n$ of degree $p^n$ ramified at the boundary. This is used to give an extrinsic description of the local behavior of a separable cover of curves in char. $p$ at a wildly ramified point whose inertia group is cyclic. In an appendix, we give an elementary computation of the conductor of such a covering, which can otherwise be determined using class field theory. | Linear systems attached to cyclic inertia | 12,147 |
Consequences of the Toda equations arising from the conjectural matrix model for the Riemann sphere are investigated. The Toda equations determine the Gromov-Witten descendent potential (including all genera) of the Riemann sphere from the degree 0 part. Degree 0 series computations via Hodge integrals then lead to higher degree predictions by the Toda equations. First, closed series forms for all 1-point invariants of all genera and degrees are given. Second, degree 1 invariants are investigated with new applications to Hodge integrals. Third, a differential equation for the generating function of the classical simple Hurwitz numbers (in all genera and degrees) is found -- the first such equation. All these results depend upon the conjectural Toda equations. Finally, proofs of the Toda equations in genus 0 and 1 are given. | The Toda equations and the Gromov-Witten theory of the Riemann sphere | 12,148 |
In this paper we consider the stratification on the moduli space of principally polarized abelian surfaces in characteristic $p>0$ defined by the height of the formal group associated to $H^2(X,O_X)$. We compute the cycle classes of the strata and compare these with those of the $p$-rank stratification. | Formal Brauer groups and the moduli of abelian surfaces | 12,149 |
In this paper we determine all finite groups G that can act on some compact Riemann surface M with the property that if H is any non-trivial subgroup of G, then the orbit surface M/H is the Riemann sphere. The idea is to look at the induced action on the vector space of holomorphic differentials on M (in the positive genus case) and then use the old-known (Wolf) classification of groups admitting fixed point-free linear actions. A description of the corresponding group actions is given in terms of Fuchsian representations. | Genus Zero Actions on Riemann Surfaces | 12,150 |
The goal of this paper is to make the vertex operator algebra approach to mirror symmetry accessible to algebraic geometers. Compared to better-known approaches using moduli spaces of stable maps and special Lagrangian fibrations, this approach follows more closely the original line of thinking that lead to the discovery of mirror symmetry by physicists. The ultimate goal of the vertex algebra approach is to give precise mathematical definitions of N=(2,2) superconformal field theories called A and B models associated to any Calabi-Yau variety and then show that thus constructed theories are related by the mirror involution for all known examples of mirror symmetric varieties. | Introduction to the vertex algebra approach to mirror symmetry | 12,151 |
The aim of this paper is to clarify and generalize techniques of works alg-geom/9711024 (see also math.AG/9810097 and math.AG/9901004). Roughly speaking, we prove that for local Fano contractions the existence of complements can be reduced to the existence of complements for lower dimensional projective Fano varieties. | The first main theorem on complements: from global to local | 12,152 |
For a finite set A of integral vectors, Gel'fand, Kapranov and Zelevinskii defined a system of differential equations with a parameter vector as a D-module, which system is called an A-hypergeometric (or a GKZ hypergeometric) system. Classifying the parameters according to the D-isomorphism classes of their corresponding A-hypergeometric systems is one of the most fundamental problems in the theory. In this paper we give a combinatorial answer for the problem under the assumption that the finite set A lies in a hyperplane off the origin, and illustrate it in two particularly simple cases: the normal case and the monomial curve case. | Isomorphism classes of A-hypergeometric systems | 12,153 |
Dropping separatedness in the definition of a toric variety, one obtains the more general notion of a toric prevariety. Toric prevarieties occur as ambient spaces in algebraic geometry and moreover they appear naturally as intermediate steps in quotient constructions. We first provide a complete description of the category of toric prevarieties in terms of convex-geometrical data, so-called systems of fans. In a second part, we consider actions of subtori H of the big torus of a toric prevariety X and investigate quotients for such actions. Using our language of systems of fans, we characterize existence of good prequotients for the action of H on X. Moreover, we show by means of an algorithmic construction that there always exists a toric prequotient for the action of H on X, that means an H-invariant toric morphism p from X to a toric prevariety Y such that every H-invariant toric morphism from X to a toric prevariety factors through p. Finally, generalizing a result of D. Cox, we prove that every toric prevariety occurs as the image of a categorical prequotient of an open toric subvariety of some complex affine space. | Toric Prevarieties and Subtorus Actions | 12,154 |
Any ample Cartier divisor D on a projective variety X is strictly nef (i.e. D.C>0 for any effective curve C on X). In general, the converse statement does not hold. But this is conjectured to be true for anticanonical divisors. The present paper establishes this fact for normal complex projective threefolds with canonical singularities. This result extends several previously known special cases. The proof rests mainly on sophisticated techniques of three dimensional birational geometry developed in the last two decades. | On the canonical threefolds with strictly nef anticanonical divisors | 12,155 |
We establish a relation between the generating functions appearing in the S-duality conjecture of Vafa and Witten and geometric Eisenstein series for Kac-Moody groups. For a pair consisting of a surface and a curve on it, we consider a refined geometric function E (involving G-bundles with parabolic structures along the curve) which depends both on elliptic and modular variables. We prove a functional equation for E with respect to the affine Weyl group, thus establishing the elliptic behavior. When the curve is P^1, we calculate the Eisenstein-Kac-Moody series explicitly and it turns out to be a certain deformation of an irreducible Kac-Moody character, more precisely, an analog of the Hall-Littlewood polynomial for the affine root system. We also get an explicit formula for the universal blowup function for any simply connected structure group. | The elliptic curve in the S-duality theory and Eisenstein series for
Kac-Moody groups | 12,156 |
We develop an intersection theory for a singular hemitian line bundle with positive curvature current on a smooth projective variety and irreducible curves on the variety. And we prove the existence of a natural rational fibration structure associated with the singular hermitian line bundle. Also for any pseudoeffective line bundle on a smooth projective variety, we prove the existence of a rational natural fibration structure associated with the line bundle. We also characterize a numerically trivial singular hermitain line bundle on a smooth projective variety. | Numerical trivial fibrations | 12,157 |
The main result of this paper is the proof of the "transversal part" of the homological mirror symmetry conjecture for an elliptic curve which states an equivalence of two $A_{\infty}$-structures on the category of vector bundles on an elliptic curves. The proof is based on the study of $A_{\infty}$-structures on the category of line bundles over an elliptic curve satisfying some natural restrictions (in particular, $m_1$ should be zero, $m_2$ should coincide with the usual composition). The key observation is that such a structure is uniquely determined up to homotopy by certain triple products. | $A_{\infty}$-structures on an elliptic curve | 12,158 |
The goal of this article is to study the equations and syzygies of embeddings of rational surfaces and certain Fano varieties. Given a rational surface X and an ample and base-point-free line bundle L on X, we give an optimal numerical criterion for L to satisfy property Np. This criterion turns out to be a characterization of property Np if X is anticanonical. We also prove syzygy results for adjunction bundles and a Reider type theorem for higher syzygies. For certain Fano varieties we also prove results on very ampleness and higher syzygies. | Some results on rational surfaces and Fano varieties | 12,159 |
It is pretty well-known that toric Fano varieties of dimension k with terminal singularities correspond to convex lattice polytopes P in R^k of positive finite volume, such that intersection of P and Z^k consists of the point 0 and vertices of P. Likewise, Q-factorial terminal toric singularities essentially correspond to lattice simplexes with no lattice points inside or on the boundary (except the vertices). There have been a lot work, especially in the last 20 years or so on classification of these objects. The main goal of this paper is to bring together these and related results, that are currently scattered in the literature. We also want to emphasize the deep similarity between the problems of classification of toric Fano varieties and classification of Q-factorial toric singularities. | Convex lattice polytopes and cones with few lattice points inside, from
a birational geometry viewpoint | 12,160 |
Working over a perfect field, I classify normal del Pezzo surfaces with base number one that contain a nonrational singularity. They form a huge infinite hierarchy; contractions of ruled surfaces lie on top of it. Descending the hierarchy hinges on a generalized version of elementary transformations. As an application, I determine the structure of 2-dimensional anticanonical models for arbitrary normal surfaces. | Normal del Pezzo surfaces containing a nonrational singularity | 12,161 |
We give some bounds on the anticanonical degrees of Fano varieties with Picard number 1 and mild singularities, extending results of Koll\'ar et al. from the early 90's and improving them even in the smooth case. The proof is based on a study of positivity properties of sheaves of differential operators on ample line bundles, and avoids the use of rational curves and bend-and-break. This note is a self-contained exposition of the main ideas of math.AG/9811022 | A new method in Fano geometry | 12,162 |
We consider subtorus actions on divisorial toric varieties. Here divisoriality means that the variety has many Cartier divisors like quasiprojective and smooth ones. We characterize when a subtorus action on such a toric variety admits a categorical quotient in the category of divisorial varieties. Our result generalizes previous statements for the quasiprojective case. An important tool for the proof is a universal reduction of an arbitrary toric variety to a divisorial one. This is done in terms of support maps, a notion generalizing support functions on a polytopal fan. A further essential step is the decomposition of a given subtorus invariant regular map to a divisorial variety into an invariant toric part followed by a non-toric part. | Quotients of Divisorial Toric Varieties | 12,163 |
We verify a special case of V. V. Shokurov's conjecture about characterization of toric varieties. More precisely, let $(X,D=\sum d_iD_i)$ be a three-dimensional log variety such that $K_X+D$ is numerically trivial and $(X,D)$ has only purely log terminal singularities. In this situation we prove the inequality \{center} $\sum d_i\le \rk\Weil(X)/(\operatorname{algebraic equivalence}) +\dim(X)$. \{center} We describe such pairs for which the equality holds and show that all of them are toric. | On a conjecture of Shokurov: Characterization of toric varieties | 12,164 |
Given a smooth projective curve $C$ of genus $g$ over the complex numbers, Torelli's thoerem asserts that the pair $(J(C),W^{g-1})$ determines $C$, where $W^{g-1}$ is an image of the $g-1$st symmetric power of $C$ inside the Jacobian under an Abel-Jacobi map. We show that the theorem holds with $g-1$ replaced by an integer $d$ in the range $1\le d\le g-1$. | A Generalized Torelli Theorem | 12,165 |
We reproduce the quantum cohomology of toric varieties (and of some hypersurfaces in projective spaces) as the cohomology of certain vertex algebras with differential. The deformation technique allows us to compute the cohomology of the chiral de Rham complex over the projective space. | Deformations of chiral algebras and quantum cohomology of toric
varieties | 12,166 |
For an essential, central hyperplane arrangement A in V=k^{n+1}, we show that \Omega^1(A) (the module of logarithmic one forms with poles along A) gives rise to a locally free sheaf on P^n if and only if for all X in L_A with rank X<dim V, the module \Omega^1(A_X) is free. Our main result is that in this case the Poicare polynomial of A is essentially the Chern polynomial. The proof is based on a result of Solomon and Terao and on a formula we give for the Chern polynomial of a bundle E on P^n in terms of the Hilbert series of \oplus_m H^0(\wedge^iE(m)). If \Omega^1(A)has projective dimension one and is locally free, we give a minimal free resolution for \Omega^p, and show that \wedge^p(\Omega^1(A))\iso\Omega^p(A), generalizing results of Rose and Terao on generic arrangements. | The Module of Logarithmic p-forms of a Locally Free Arrangement | 12,167 |
A line bundle with a base-point-free multiple is called semiample. I give a cohomological characterization of semiample line bundles. The result is a common generalization of the Fujita-Zariski criterion for semiampleness and the Grothendieck-Serre characterization of ampleness. Applying the Fujita-Zariski Theorem, I characterize contractible curves in 1-dimensional families. | A characterization of semiampleness and contractions of relative curves | 12,168 |
Let X be a complex surface with no nontrivial 2-forms. Then we show that Bloch's conjecture is true (i.e. the Albanese map in this case is injective) if and only if any homologically trivial idempotent in the ring of correspondences vanishes. Furthermore the cube of the ideal of homologically trivial correspondences is zero if these equivalent conditions are satisfied (e.g. if X is not of general type). | Bloch's Conjecture and Chow Motives | 12,169 |
In this paper we approach the study of generalized theta linear series on moduli of vector bundles on curves via vector bundle techniques on abelian varieties. We study what are called the Verlinde bundles in order to obtain information about duality between theta functions and effective global and normal generation on these moduli spaces. | Verlinde bundles and generalized theta linear series | 12,170 |
We obtain effective results for the global generation of pluritheta line bundles on moduli spaces of vector bundles on curves. The main ingredient is an independent result giving an upper bound on the dimension of the Hilbert scheme of coherent quotients of a given vector bundle. | Dimension estimates for Hilbert schemes and effective base point
freeness on moduli spaces of vector bundles on curves | 12,171 |
We continue our work on variations of graded-polarized mixed Hodge structures by defining analogs of the harmonic metric equations for filtered bundles and proving a precise analog of Schmid's Nilpotent Orbit Theorem for 1-parameter degenerations of graded-polarized mixed Hodge structure. | Degenerations of Mixed Hodge structure | 12,172 |
We study the existence of $L^2$ holomorphic sections of invariant line bundles over Galois coverings of Zariski open sets in Moishezon manilolds. We show that the von Neuman dimension of the space of $L^2$ holomorphic sections is bounded below under reasonable curvature conditions. We also give criteria for a a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. Their coverings are then studied with the same methods. As applications we give weak Lefschetz theorems using the Napier--Ramachandran proof of the Nori theorem. | L^2-Riemann-Roch Inequalities for Covering Manifolds | 12,173 |
In this note we study the moduli space of rank two semistable sheaves on a smooth cubic hypersurface $X\subset\mathbb{P}^{4}$ with $c_{1}=0$, $c_{2}=2$ and $c_{3}=0$. We show that it is isomorphic to the blow-up of the intermediate jacobian $J(X)$ of $X$ along the Fano surface of $X$. We complete previous results of Iliev-Markushevich AG/9910058 and Markushevich-Tikhomirov AG/9910063. | Espace des modules des faisceaux semi-stables de rang 2 et de classes de
Chern $c_{1}=0$, $c_{2}=2$ et $c_{3}=0$ sur une hypersurface cubique lisse de
$\mathbb{P}^{4}$ | 12,174 |
The moduli space of cubic surfaces in complex projective space is known to be isomorphic to the quotient of the complex 4-ball by a certain arithmetic group. We apply Borcherds' techniques to construct automorphic forms for this group and show that these provide an embedding of the moduli space in 9-dimensional projective space. We also show that our automorphic forms directly encode the geometry of cubic surfaces, by showing that each of Cayley's invariants (certain cross-ratios) is simply a quotient of two of our automorphic forms. | Cubic Surfaces and Borcherds Products | 12,175 |
For a semistable reflexive sheaf $E$ of rank $r$ and $c_1=a$ on $\P^n$ and an integer $d$ such that $r|ad$, we give sufficient conditions so that the restriction of $E$ on a generic rational curve of degree $d$ is balanced, i.e. a twist of the trivial bundle (for instance, if $E$ has balanced restriction on a generic line, or $r=2$ or $E$ is an exterior power of the tangent bundle). Assuming this, we give a formula for the 'virtual degree', interpreted enumeratively, of the locus of rational curves of degree $d$ on which the restriction of $E$ is not balanced, generalizing a classical formula due to Barth for the degree of the divisor of jumping lines of a semistable rank-2 bundle. | The degree of the divisor of jumping rational curves | 12,176 |
An explicit projective embedding of the moduli space of marked cubic surfaces is given. This embedding is equivariant under the Weyl group of type E6. The image is defined by a system of linear and cubic equations. To express the embedding in a most symmetric way, the target would be 79-dimensional, however the image lies in a 9-dimensional linear subspace. | A W(E_6)-equivariant projective embedding of the moduli space of cubic
surfaces | 12,177 |
We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal surface (degree, class, class of its hyperplane section, sectional genus and degrees of the nodal and cuspidal curve). We study in particular the congruences of chords to a smooth curve and the congruences of bitangents or flexes to a smooth surface. We find that they possess unexpected components in their focal surface, and conjecture that they are the only ones with this property. | A focus on focal surfaces | 12,178 |
We describe a new perspective on the intersection theory of the moduli space of curves involving both Virasoro constraints and Gorenstein conditions. The main result of the paper is the computation of a basic 1-point Hodge integral series occurring in the tautological ring of the moduli space of nonsingular curves. In the appendix by D. Zagier, "Polynomials arising from the tautological ring", a detailed study is made of certain polynomials whose coefficients are intersection numbers on moduli space. The paper and the appendix together provide proofs of all previously conjectured formulas for 1-point integrals in the tautological ring, and of natural extensions of these formulas as well. | Logarithmic series and Hodge integrals in the tautological ring (with an
appendix by D. Zagier) | 12,179 |
Ran proved that smooth codimension 2 varieties in ${\bf P}^{m+2}$ are $j$-normal if $(j+1)(3j-1)\le m-1$, in this paper we extend this result to small codimension projective varieties. Let $X$ be a r codimension subvariety of $\pro$, we prove that if the set $\Sigma_{(j+1)}$ of $(j+1)$-secants to $X$ through a generic external point is not empty, $2(r+1)j\leq m-r$ and $(j+1)((r+1)j-1)\leq m-1$ then $X$ is $j$-normal. If $X$ is given by the zero locus of a section of a rank $r$ vector bundle $E$ on $\pro$, we prove that $\textrm{deg} \Sigma_{j+1}=\frac{1}{(j+1)!}\prod_{i=0}^{j}c_{r}(E(-i))$. Moreover we get a new simple proof of Zak's theorem on linear normality if $m\ge 3r$. Finally we prove that if $c_{r}(N(-2))\neq 0$ and $6r\le m-4$ then $X$ is 2-normal. | On quadratic and higher normality of small codimension projective
varieties | 12,180 |
We study the general elements of the moduli spaces (\MM_{\PP^2} (r, c_1, c_2) ) of stable holomorphic vector bundle on $\PP^2$ and their minimal free resolution. Incidentally, a quite easy proof of the irreducibility of (\MM_{\PP^2} (r, c_1, c_2)) is shown. | Minimal resolution of general stable vector bundles on $\PP^2$ | 12,181 |
We generalize results of the paper math.AG/9803144, in which Chisini's conjecture on the unique reconstruction of f by the curve B is investigated. For this fibre products of generic coverings are studied. The main inequality bounding the degree of a covering in the case of existence of two nonequivalent coverings with the branch curve B is obtained. This inequality is used for the proof of the Chisini conjecture for m-canonical coverings of surfaces of general type for $m\ge 5$. | Generic coverings of plane with A-D-E-singularities | 12,182 |
The real intersection cohomology of a toric variety is described in a purely combinatorial way using methods of elementary commutative algebra only. We define, for arbitrary fans, the notion of a ``minimal extension sheaf'' on the fan as an axiomatic characterization of the equivariant intersection cohomology sheaf. This provides a purely algebraic interpretation of Stanley's generalized f- and g-vector of an arbitrary polytope or complete fan under a natural vanishing condition. -- The results presented in this note originate from joint work with G.Barthel, J.-P.Brasselet and L.Kaup, continuing earlier research (see math.AG/9904159). A detailed exposition will appear elsewhere (see math.AG/0002181). | Towards a combinatorial Intersection Cohomology for Fans | 12,183 |
Let Y be a projective non-singular curve of genus g, X a projective manifold, both defined over the field of complex numbers, and let f:X ---> Y be a surjective morphism with general fibre F. If the Kodaira dimension of X is non-negative, and if Y is the projective line we show that f has at least 3 singular fibres. In general, for non-isotrivial morphisms f, one expects that the number of singular fibres is at least 3, if g=0, or at least 1, if g=1. Using the strong additivity of the Kodaira dimension, this is verified, if either F is of general type, or if F has a minimal model with a semi-ample canonical divisor. The corresponding result has been obtained by Migliorini and Kovacs, for families of surfaces of general type and for families of canonically polarized manifolds, and by Oguiso-Viehweg for families of elliptic surfaces. As a byproduct we obtain explicit bounds for the degree of the direct image of powers of the dualizing sheaf, generalizing those obtained by Bedulev-Viehweg for families of surfaces of general type. | On the isotriviality of families of projective manifolds over curves | 12,184 |
The aim of this note is to study local and global Seshadri constants for a family of smooth surfaces with prescribed polarization. We shall first observe that given $\alpha$ being smaller than the square root of the degree of polarization, the set of local Seshadri constants in the range $(0, \alpha]$ is finite. This in particular implies that the square root of the degree of polarization is the only possible accumulation point of the set of local Seshadri constants. Next we shall remark the Zariski closedness of the set of points whose local Seshadri constants are in any given interval $(0, a]$. As applications, we shall also add a few remarks on the lower semi-continuity of both local and global Seshadri constants with respect to parameters involved, and on the minimality and the maximality of their infimum and supremum. | Seshadri constant for a family of surfaces | 12,185 |
We show that for a polynomial map, the size of the Jordan blocks for the eigenvalue 1 of the monodromy at infinity is bounded by the multiplicity of the reduced divisor at infinity of a good compactification of a general fiber. The existence of such Jordan blocks is related to global invariant cycles of the graded pieces of the weight filtration. These imply some applications to period integrals. We also show that such a Jordan block of size greater than 1 for the graded pieces of the weight filtration is the restriction of a strictly larger Jordan block for the total cohomology group. If there are no singularities at infinity, we have a more precise statement on the monodromy. | Monodromy at Infinity and the Weights of Cohomology | 12,186 |
We calculate Tits buildings for certain arithmetic subgroups of Sp(4). These give information about the boundary of the corresponding moduli spaces of abelian surfaces. More pictures (in colour) and a summary of the results (in English) can be found at http://www.bath.ac.uk/~masgks/Buildings/ | Das Titsgebaeude von Siegelschen Modulgruppen vom Geschlecht 2 | 12,187 |
In 1932 F. Severi claimed, with an incorrect proof, that every smooth minimal projective surface $S$ such that the bundle $\Omega^1_S$ is generically generated by global sections satisfies the topological inequality $2c_1^2(S)\ge c_2(S)$. According to Enriques-Kodaira classification, the above inequality is easily verified when the Kodaira dimension of the surface is $\le 1$, while for surfaces of general type it is still an open problem known as Severi conjecture. In this paper we prove Severi conjecture under the additional mild hypothesis that $S$ has ample canonical bundle. Moreover, under the same assumption, we prove that $2c_1^2(S)=c_2(S)$ if and only if $S$ is a double cover of an abelian surface. | Surfaces of Albanese general type and the Severi Conjecture | 12,188 |
Let V be a finite dimensional vector space over a local field. Let us say that a complex function on V is elementary if it is a product of the additive character of a rational function Q on V and multiplicative characters of polynomials on V. In this paper we study when the Fourier transform of an elementary function is elementary. If Q has a nonzero Hessian, a necessary condition for this is that the Legendre transform Q_* of Q is rational. The basic example is a nondegenerate quadratic form. We study such functions Q, give examples, and find all of them such that both Q and Q_* are of the form f(x)/t, where f is a cubic form in many variables (the simplest case after quadratic forms). It turns out that this classification is closely related to Zak's classification of Severi varieties. The second half of the paper is devoted to finding and classifying elementary functions with elementary Fourier transforms when Q is a fixed function with rational Q_*. We consider the simplest case when Q is a monomial, and classify combinations of multiplicative characters that can arise. The answer (for real and complex fields) is given in terms of exact covering systems. We also describe examples related to prehomogeneous vector spaces. Finally, we consider examples over p-adic fields, and in particular give a local proof of an integral formula of D.K. that could previously be proved only by a global method. | When is the Fourier transform of an elementary function elementary? | 12,189 |
Let ${\rm F}$ be a rank-2 semi-stable sheaf on the projective plane, with Chern classes $c_{1}=0,c_{2}=n$. The curve $\beta_{\rm F}$ of jumping lines of ${\rm F}$, in the dual projective plane, has degree $n$. Let ${\rm M}_{n}$ be the moduli space of equivalence classes of semi-stables sheaves of rank 2 and Chern classes $(0,n)$ on the projective plane and ${\cal C}_{n}$ be the projective space of curves of degree $n$ in the dual projective plane. The Barth morphism $$\beta: {\rm M}_{n}\longrightarrow{\cal C}_{n}$$ associates the point $\beta_{\rm F}$ to the class of the sheaf ${\rm F}$. We prove that this morphism is generically injective for $n\geq 4.$ The image of $\beta$ is a closed subvariety of dimension $4n-3$ of ${\cal C}_{n}$; as a consequence of our result, the degree of this image is given by the Donaldson number of index $4n-3$ of the projective plane. | Sur le morphisme de Barth | 12,190 |
The article has two parts. The first part is devoted to proving a singular version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem of Esnault and Viehweg. This is then used to prove other vanishing theorems. In the second part these vanishing theorems are used to prove an Arakelov-Parshin type boundedness result for families of canonically polarized varieties with rational Gorenstein singularities. | Logarithmic Kodaira-Akizuki-Nakano vanishing and Arakelov-Parshin
boundedness for singular varieties | 12,191 |
Let X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain 'motivic integral', living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the motivic (Igusa) zeta function, associated to a regular function on X, which specializes to both the classical p-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant. This paper treats a generalization to singular varieties. Batyrev already considered such a 'Kontsevich invariant' for log terminal varieties (on the level of Hodge polynomials instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any Q-Gorenstein variety X we associate a motivic zeta function and a 'Kontsevich invariant' to effective Q-Cartier divisors on X whose support contains the singular locus of X. | Zeta functions and 'Kontsevich invariants' on singular varieties | 12,192 |
Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula expressing Hurwitz numbers (counting covers of the projective line with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual localization on the moduli space of stable maps, and describe how the proof could be simplified by the proper algebro-geometric definition of a "relative space". | Hodge integrals and Hurwitz numbers via virtual localization | 12,193 |
In this paper we give the projective generation of congruences of order 1 of r-dimensional projective spaces in P^N from their focal loci. In a natural way, this construction shows that the corresponding surfaces in the grassmannian are the Veronese surface, and rational ruled surfaces eventually with singularities. We characterize when these surfaces are smooth, recovering and generalizing a Ziv Ran's result. | Projective generation and smoothness of congruences of order 1 | 12,194 |
This paper gives the all possible global indices of log Calabi-Yau 3-folds with standard coefficients on the boundaries and having lc, non-klt singularities. This follows easily from the discussion in the paper: The indices of log canonical singularities by Fujino (AG/9909035). | The global indices of log Calabi-Yau varieties --A supplement to
Fujino's paper: The indices of log canonical singularities-- | 12,195 |
We study the following question: fix a sufficient general curve D of degree d in P^2, what is the least number of intersections between D and an irreducible curve of degree m? G. Xu proved this number i(d, m) is at least d - 2 for all m. This problem can be regarded as the algebraic part of Kobayashi conjecture on the hyperbolicity of P^2 D. We first improved Xu's bound with m fixed and then generalized his result to rational ruled surfaces. | On the Intersection of Two Plane Curves | 12,196 |
Our main theorem characterizes the complete intersections of codimension 2 in a projective space of dimension 3 or more over an algebraically closed field of characteristic 0 as the subcanonical and self-linked subschemes. In order to prove this theorem, we'll prove the Gherardelli linkage theorem, which asserts that a partial intersection of two hypersurfaces is subcanonical if and only if its residual intersection is, scheme-theoretically, the intersection of the two hypersurfaces with a third. | Gherardelli linkage and complete intersections | 12,197 |
We apply the homological mirror symmetry for elliptic curves to the study of indefinite theta series. We prove that every such series corresponding to a quadratic form of signature (1,1) can be expressed in terms of theta series associated with split quadratic forms and the usual theta series. We also show that indefinite theta series corresponding to univalued Massey products between line bundles on elliptic curve are modular. | Indefinite theta series of signature (1,1) from the point of view of
homological mirror symmetry | 12,198 |
Hyperquot schemes are generalizations of Grothendieck's Quot scheme to partial flags. Using a Bialynicki-Birula decomposition, we obtain combinatorial data for the Betti numbers, and collect this information into the form of rational generating functions for the Poincare polynomials. | Poincare polynomials of hyperquot schemes | 12,199 |
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