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We study a kind of modification of an affine domain which produces another affine domain. First appeared in passing in the basic paper of O. Zariski (1942), it was further considered by E.D. Davis (1967). The first named author applied its geometric counterpart to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions which guarantee preservation of the topology under a modification. As an application, we show that the group of biregular automorphisms of the affine hypersurface $X \subset C^{k+2}$ given by the equation $uv=p(x_1,...,x_k)$ where $p \in C[x_1,...,x_k],$ acts $m-$transitively on the smooth part reg$X$ of $X$ for any $m \in N.$ We present examples of such hypersurfaces diffeomorphic to Euclidean spaces. | Affine modifications and affine hypersurfaces with a very transitive
automorphism group | 11,700 |
In the paper we show that for a normal-crossings degeneration $Z$ over the ring of integers of a local field with $X$ as generic fibre, the local monodromy operator and its powers determine invariant cocycle classes under the decomposition group in the cohomology of the product $X \times X$. More precisely, they also define algebraic cycles on the special fibre of a resolution of $Z \times Z$. In the paper, we give an explicit description of these cycles for a degeneration with at worst triple points as singularities. These cycles explain geometrically the presence of poles on specific local factors of the L-function related to $X \times X$. | The local monodromy as a generalized algebraic correspondence | 11,701 |
The rank 4 locus of a general skew-symmetric 7x7 matrix gives the pfaffian variety in P^20 which is not defined as a complete intersection. Intersecting this with a general P^6 gives a Calabi-Yau manifold. An orbifold construction seems to give the 1-parameter mirror-family of this. However, corresponding to two points in the 1-parameter family of complex structures, both with maximally unipotent monodromy, are two different mirror-maps: one corresponding to the general pfaffian section, the other to a general intersection of G(2,7) in P^20 with a P^13. Apparently, the pfaffian and G(2,7) sections constitute different parts of the A-model (Kahler structure related) moduli space, and, thus, represent different parts of the same conformal field theory moduli space. | The Pfaffian Calabi-Yau, its Mirror, and their link to the Grassmannian
G(2,7) | 11,702 |
We use the notion of Milnor fibres of the germ of a meromorphic function and the method of partial resolutions for a study of topology of a polynomial map at infinity (mainly for calculation of the zeta-function of a monodromy). It gives effective methods of computation of the zeta-function for a number of cases and a criterium for a value to be atypical at infinity. | On the zeta-function of a polynomial at infinity | 11,703 |
We give a new formula for the Chern-Schwartz-MacPherson class of a hypersurface in a nonsigular compact complex analytic variety. In particular this formula generalizes our previous result on the Euler characteristic of such a hypersurface. Two different approaches are presented. The first is based on the theory of characteristic cycle and the works of Sabbah, Briancon-Maisonobe-Merle, and Le-Mebkhout. In particular, this approach leads to a simple proof of a formula of Aluffi for the above mentioned class. The second approach uses Verdier's specialization property of the Chern-Schwartz-MacPherson classes. Some related new formulas are also given. | Characteristic Classes of Hypersurfaces and Characteristic Cycles | 11,704 |
We prove a conjecture of Frenkel-Gaitsgory-Kazhdan-Vilonen on some exponential sums related to the geometric Langlands correspondence. Our main ingredients are the resolution of Lusztig scheme of lattices introduced by Laumon and the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber. | Preuve d'une conjecture de Frenkel-Gaitsgory-Kazhdan-Vilonen | 11,705 |
The variety of sums of powers of a homogeneous polynomial of degree d in n variables is defined and investigated in some examples, old and new. These varieties are studied via apolarity and syzygies. Classical results of Sylvester (1851), Hilbert (1888), Dixon and Stuart (1906) and some more recent results of Mukai (1992) are presented together with new results for the cases (n,d)=(3,8), (4,2), (5,3). In the last case the variety of sums of 8 powers of a general cubic form is a Fano 5-fold of index 1 and degree 660. | Varieties of sums of powers | 11,706 |
Let $X$ be a complete algebraic variety over {\bf C}. We consider a log variety $(X,\Delta)$ that is weakly Kawamata log terminal. We assume that $K_X+\Delta$ is a {\bf Q}-Cartier {\bf Q}-divisor and that every irreducible component of $\lfloor \Delta \rfloor$ is {\bf Q}-Cartier. A nef and big Cartier divisor $H$ on $X$ is called {\it nef and log big} on $(X,\Delta)$ if $H |_B$ is nef and big for every center $B$ of non-"Kawamata log terminal" singularities for $(X,\Delta)$. We prove that, if $L$ is a nef Cartier divisor such that $aL-(K_X+\Delta)$ is nef and log big on $(X,\Delta)$ for some $a \in$ {\bf N}, then the complete linear system $| mL |$ is base point free for $m \gg 0$. | A Base Point Free Theorem of Reid Type, II | 11,707 |
We give an algorithm to compute the following cohomology groups on $U = \C^n \setminus V(f)$ for any non-zero polynomial $f \in \Q[x_1, ..., x_n]$; 1. $H^k(U, \C_U)$, $\C_U$ is the constant sheaf on $U$ with stalk $\C$. 2. $H^k(U, \Vsc)$, $\Vsc$ is a locally constant sheaf of rank 1 on $U$. We also give partial results on computation of cohomology groups on $U$ for a locally constant sheaf of general rank and on computation of $H^k(\C^n \setminus Z, \C)$ where $Z$ is a general algebraic set. Our algorithm is based on computations of Gr\"obner bases in the ring of differential operators with polynomial coefficients. | An algorithm for de Rham cohomology groups of the complement of an
affine variety via D-module computation | 11,708 |
This article recounts the interaction of topology and singularity theory (mainly singularities of complex algebraic varieties) which started in the early part of this century and bloomed in the 1960's with the work of Hirzebruch, Brieskorn, Milnor and others. Some of the topics are followed to the present day. (A chapter for the book "History of Topology", ed. I. M. James) | Singularities | 11,709 |
We characterize the vector bundles on G(1,4) that have no intermediate cohomology. We obtain them from extensions of the universal bundles and others related with them. In particular, we get a characterization of the universal vector bundles from their cohomology. | Vector bundles on G(1,4) without intermediate cohomology | 11,710 |
The aim of this note is a combinatorial description of a category of $D$-modules over an affine space, smooth along the stratification defined by an arrangement of hyperplanes. These $D$-modules are assumed to satisfy certain non-resonance condition. The main result, see Theorem 4.1, generalizes [S.Khoroshkin, $D$-modules over arrangements of hyperplanes, Comm. in Alg. 23(9) (1995), 3481-3504]. | Non-resonance D-modules over arrangements of hyperplanes | 11,711 |
This paper aims at generalizing some geometric properties of Grassmannians of finite dimensional vector spaces to the case of Grassmannnians of infinite dimensional ones, in particular for that of $k((z))$. It is shown that the Determinant Line Bundle generates its Picard Group and that the Pl\"ucker equations define it as closed subscheme of a infinite projective space. Finally, a characterization of finite dimensional projective spaces in Grassmannians allows us to offer an approach to the study of the automorphism group. | Grassmannian of $k((z))$: Picard Group, Equations and Automorphisms | 11,712 |
This note consists of two parts. Part I is an exposition of (a part of) the V.Drinfeld's letter, [D]. The sheaf of algebras of polyvector fields on a Calabi-Yau manifold, equipped with the Schouten bracket, admits a structure of a Batalin-Vilkovisky algebra. This fact was probably first noticed by Z.Ran, [R]. Part II is devoted to some generalizations of this remark. | Remarks on formal deformations and Batalin-Vilkovisky algebras | 11,713 |
The Severi variety V_{n,d} of a smooth projective surface S is defined as the subvariety of the linear system |O_S(n)|, which parametrizes curves with d nodes. We show that, for a general surface S of degree k in P^3 and for all n>k-1, d=0,...,dim(|O_S(n)|), there exists one component of V_{n,d} which is reduced, of the expected dimension dim(|O_S(n)|)-d. Components of the expected dimension are the easiest to handle, trying to settle an enumerative geometry for singular curves on surfaces. On the other hand, we also construct examples of reducible Severi varieties, on general surfaces of degree k>7 in P^3. | On the Severi varieties of surfaces in P^3 | 11,714 |
Castelnuovo-Mumford regularity is an important invariant of projective algebraic varieties. A well known conjecture due to Eisenbud and Goto gives a bound for regularity in terms of the codimension and degree. This conjecture is known to be true for curves (Gruson-Lazarsfeld-Peskine) and smooth surfaces (Pinkham, Lazarsfeld), but not in general. The purpose of this paper is to give new bounds for the regularity of smooth varieties in dimensions 3 and 4 that are only slightly worse than the optimal ones suggested by the conjecture. Our method yields new bounds up to dimension 14, but as they get progressively worse for higher dimensions, we have not written them down here. | Castelnuovo regularity for smooth projective varieties of dimensions 3
and 4 | 11,715 |
Castelnuovo-Mumford regularity is an important invariant of projective algebraic varieties. A well known conjecture due to Eisenbud and Goto gives a bound for regularity in terms of the codimension and degree,i.e., Castelnuovo-Mumford regularity of a given variety $X$ is less than or equal to $deg(X)-codim(X)+1$. This regularity conjecture (including classification of examples on the boundary) was verified for integral curves (Castelnuovo, Gruson, Lazarsfeld and Peskine), and for smooth surfaces (Pinkham, Lazarsfeld). In this paper we prove that $reg(X) \le deg(X)-1$ for smooth threefolds $X$ in P^5 and that the only varieties on the boundary are the Segre threefold and the complete intersection of two quadrics. Furthermore, every smooth threefold $X$ in P^5 is $k$-normal for all $k \ge deg(X)-4$, which is the optimal bound as the Palatini 3-fold of degree 7 shows. | Castelnuovo-Mumford Regularity of Smoth Threefolds in P^5 | 11,716 |
Let X be a projective irreducible smooth algebraic variety. A "fine moduli space" of sheaves on X is a family F of coherent sheaves on X parametrized by an integral variety M such that : F is flat on M; for all distinct points x, y of M the sheaves F_x, F_y on X are not isomorphic and F is a complete deformation of F_x; F has an obvious local universal property. We define also a "fine moduli space defined locally", where F is replaced by a family (F_i), where F_i is defined on an open subset U_i of M, the U_i covering M. This paper is devoted to the study of such fine moduli spaces. We first give some general results, and apply them in three cases on the projective plane : the fine moduli spaces of prioritary sheaves, the fine moduli spaces consisting of simple rank 1 sheaves, and those which come from moduli spaces of morphisms. In the first case we give an exemple of a fine moduli space defined locally but not globally, in the second an exemple of a maximal non projective fine moduli space, and in the third we find a projective fine moduli space consisting of simple torsion free sheaves, containing stable sheaves, but which is different from the corresponding moduli space of stable sheaves. | Varietes de modules alternatives | 11,717 |
We describe the Hilbert scheme components parametrizing lines and conics on the space of determinantal nets of conics, N. As an application, we use the quantum Lefschetz hyperplane principle to compute the instanton numbers of rational curves on a complete intersection Calabi-Yau threefold in N. We also compute the number of lines and conics on some Calabi-Yau sections of non-decomposable vector bundles on N. The paper contains a brief summary of the A-model theory leading up to Givental-Kim's quantum Lefschetz hyperplane principle. | Rational Curves on the Space of Determinantal Nets of Conics | 11,718 |
We develop an approach to noncommutative algebraic geometry ``in the perturbative regime" around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal neighborhood of M=Spec(A) in the (nonexistent) space Spec(R). This is a ringed space (M,O) where O is a certain sheaf of noncommutative rings on M. Such ringed spaces can be glued together to form more global objects called NC-schemes. We are especially interested in NC-manifolds, NC-schemes for which the completion of O at every point of M is isomorphic to the algebra of noncommutative power series (completion of the free associative algebra). An explicit description of the simplest NC-manifold, the affine space, is given by using the Feynman-Maslov calculus of ordered operators. We show that many familiar algebraic varieties can be naturally enlarged to NC-manifolds. Among these are all the classical flag varieties and all the smooth moduli spaces of vector bundles. | Noncommutative geometry based on commutator expansions | 11,719 |
We shall investigate index 1 covers of 2-dimensional log terminal singularities. The main result is that the index 1 cover is canonical if the characteristic of the base field is different from 2 or 3. We also give some counterexamples in the case of characteristic 2 or 3. By using this result, we correct an error in a previous paper. | Index 1 covers of log terminal surface sigularities | 11,720 |
In this paper, we prove the l-independence of monodromy weight filtration for a geometrically smooth variety over an equicharacteristic local field. We also prove the l-independence for the geometric monodromy representation on the associated graded module of weight monodromy filtration. | Monodromy weight filtration is independent of l | 11,721 |
This is the third of a series of papers studying real algebraic threefolds, but the methods are mostly independent from the previous two. Let $f:X\to S$ be a map of a smooth projective real algebraic 3-fold to a surface $S$ whose general fibers are rational curves. Assume that the set of real points of $X$ is an orientable 3-manifold $M$. The aim of the paper is to give a topological description of $M$. It is shown that $M$ is either Seifert fibered or a connected sum of lens spaces. Much stronger results hold if $S$ is rational. | Real Algebraic Threefolds III: Conic Bundles | 11,722 |
In this paper, we will describe the mathematical foundation of topological Landau-Ginzburg (LG) models coupled to gravity at genus 0 in terms of primitive forms. We also discuss the mirror symmetry for Calabi-Yau manifolds and CP^1 in our context. We will show that the mirror partner of CP^1 is the theory of primitive form associated to f=z+qz^{-1}. | Primitive Forms, Topological LG models coupled to Gravity and Mirror
Symmetry | 11,723 |
This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. It is based on lectures given at Montr\'eal is Summer 1997. Our main aim is to obtain explicit descriptions of cohomology or Chow rings of certain manifolds with group actions which arise in representation theory, e.g. homogeneous spaces and their compactifications. As another appplication of equivariant intersection theory, we obtain simple versions of criteria for smoothness or rational smoothness of Schubert varieties (due to Kumar, Carrell-Peterson and Arabia) whose statements and proofs become quite transparent in this framework. | Equivariant cohomology and equivariant intersection theory | 11,724 |
We study the geometry of algebraic monoids. We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid. Moreover, if this group is reductive, then the monoid is affine. We then give a combinatorial classification of reductive monoids by means of the theory of spherical varieties. | Algebraic monoids and group embeddings | 11,725 |
In characteristic zero, semistable principal bundles on a nonsingular projective curve with a semisimple structure group form a bounded family, as shown by Ramanathan in 1970's using the Narasimhan-Seshadri theorem. This was the first step in his construction of moduli for principal bundles. In this paper we prove boundedness in finite characteristics (other than characteristic 2), when the structure group is a semisimple, simply connected algebraic group of classical type. The main ingredient is an analogue of the Mukai-Sakai theorem (which says that any vector bundle admits a proper subbundle whose degree is `not too small') in the present situation. | Boundedness of semistable principal bundles on a curve, with classical
semisimple structure groups | 11,726 |
We define the Chow ring of the classifying space of a linear algebraic group. In all the examples where we can compute it, such as the symmetric groups and the orthogonal groups, it is isomorphic to a natural quotient of the complex cobordism ring of the classifying space, a topological invariant. We apply this to get torsion information on the Chow groups of varieties defined as quotients by finite groups. This generalizes Atiyah and Hirzebruch's use of such varieties to give counterexamples to the Hodge conjecture with integer coefficients. | The Chow ring of a classifying space | 11,727 |
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied is g_2=(r-1)(r-3)/8, and for this genus there are two non-isomorphism maximal curves known when r \equiv 3 (mod 4). Here, a maximal curve with genus g_2 and a non-singular plane model is characterized as a Fermat curve of degree (r+1)/2. | On plane maximal curves | 11,728 |
We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the number of elliptic (and rational) plane curves with various "codimension 1" behavior (cuspidal, tacnodal, triple pointed, etc., as well as the geometric and arithmetic sectional genus of the Severi variety). We compute the latter numbers for genus 2 and 3 plane curves as well. We rely on results of Caporaso, Diaz, Getzler, Harris, Ran, and especially Pandharipande. | Enumerative geometry of plane curves of low genus | 11,729 |
In the present paper Mori extremal rays of a smooth projective manifold X are divided into two classes: L-supported and L-negligible (where ``L'' stands for ``Lefschetz'' since the division comes from Hard Lefschetz Theorem). Roughly speaking: L-supported rays are strongly distinguishable in topology while L-negligible rays have very mild geometry. Each L-supported ray R defines hyperplane in H^2(X,R) on which Lefschetz duality degenerates so it is a cohomology ring invariant. The hyperplane carries a multiplicity (cohomology ring invariant) which is related to the geometry of the ray R. The number of L-supported rays is bounded. Although the number of L-negligible rays may be infinite and they are invisible in the cohomology ring, their geometry is easier than that of L-supported rays. They are classifieable in low dimensions. Each L-negligible ray contains lots of ``good'' rational curves whose deformation is of expected dimension. In effect, L-negligible rays are invariant under deformations of complex structure and can be used to compute Gromov-Witten invariants in symplectic geometry. | Cohomological invariants of complex manifolds coming from extremal rays | 11,730 |
We study several subseries of the space of second order theta functions on the Jacobian of a non-hyperelliptic curve. In particular, we are interested in the subseries P\Gamma_{00} consisting of 2theta-divisors having multiplicity at least 4 at the origin, or, equivalently, containing the surface C-C, and in its analogues consisting of 2theta-divisors having higher multiplicities at the origin, containing the four-fold Sym^2C-Sym^2C, or singular along the surface C-C. We use rank 2 vector bundles with a given number of global sections to prove canonical isomorphisms between quotients of the above introduced subseries and vector spaces defined by the canonical divisor. | Singularities of 2theta-divisors in the Jacobian | 11,731 |
This note is a continuation of our paper with E. Zaslow "Categorical mirror symmetry: the elliptic curve", math.AG/9801119. We compare some triple Massey products on elliptic curve with the corresponding Fukaya products on the symplectic torus and recover the classical identity due to Kronecker. We also compute some triple Fukaya products such that the corresponding Massey products are not correctly defined. | Massey and Fukaya products on elliptic curves | 11,732 |
This paper addresses the following question of Oort: "Are there any postive dimensional locally symmetric subvarieties of the moduli space of abelian varieties that are contained in the jacobian locus and contain the jacobian of at least one smooth curve? Some partial results are given. | Locally symmetric families of curves and jacobians | 11,733 |
We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series associated to p-adic points on a p-adic variety. The main tools which are used are semi-algebraic geometry in spaces of power series and motivic integration (a notion introduced by M. Kontsevich). In particular we develop the theory of motivic integration for semi-algebraic sets of formal arcs on singular algebraic varieties, we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of Oesterle. | Germs of arcs on singular algebraic varieties and motivic integration | 11,734 |
We define motivic analogues of Igusa's local zeta functions. These functions take their values in a Grothendieck group of Chow motives. They specialize to p-adic Igusa local zeta functions and to the topological zeta functions we introduced several years ago. We study their basic properties, such as functional equations, and their relation with motivic nearby cycles. In particular the Hodge spectrum of a singular point of a function may be recovered from the Hodge realization of these zeta functions. | Motivic Igusa zeta functions | 11,735 |
The aim of this note is to define certain sheaves of vertex algebras on smooth manifolds. For each smooth complex algebraic (or analytic) manifold $X$, we construct a sheaf $\Omega^{ch}_X$, called the {\bf chiral de Rham complex} of $X$. It is a sheaf of vertex algebras in the Zarisky (or classical) topology, It comes equipped with a $\BZ$-grading by {\it fermionic charge}, and the {\it chiral de Rham differential} $d_{DR}^{ch}$, which is an endomorphism of degree 1 such that $(d_{DR}^{ch})^2=0$. One has a canonical embedding of the usual de Rham complex $(\Omega_X, d_{DR})\hra (\Omega_X^{ch}, d_{DR}^{ch})$ which is a quasiisomorphism. If $X$ is Calabi-Yau then this sheaf admits an N=2 supersymmetry. For some $X$ (for example, for curves or for the flag spaces $G/B$), one can construct also a purely even analogue of this sheaf, a {\it chiral structure sheaf} $\CO^{ch}_X$. For the projective line, the space of global sections of the last sheaf is the irreducible vacuum $\hsl(2)$-module on the critical level. | Chiral de Rham complex | 11,736 |
In this paper we show the lower bound of the set of non-zero $-K^2$ for normal surface singularities establishing that this set has no accumulation points from above. We also prove that every accumulation point from below is a rational number and every positive integer is an accumulation point. Every rational number can be an accumulation point modulo $\bZ$. We determine all accumulation points in $[0, 1]$. If we fix the value $-K^2$, then the values of $p_g$, $p_a$, mult, embdim and the numerical indices are bounded, while the numbers of the exceptional curves are not bounded. | On $-K^2$ for normal surface singularities | 11,737 |
We introduce motivic analogues of p-adic exponential integrals. We prove a basic multiplicativity property from which we deduce a motivic analogue of the Thom-Sebastiani Theorem. In particular, we obtain a new proof of the Thom-Sebastiani Theorem for the Hodge spectrum of (non isolated) singularities of functions. | Motivic exponential integrals and a motivic Thom-Sebastiani Theorem | 11,738 |
A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic terms of semi-simple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torus-equivariant Gromov - Witten invariants of compact K\"ahler manifolds with isolated fixed points and for concave bundle spaces over such manifolds. Several results on genus 0 Gromov - Witten theory include: a non-linear Serre duality theorem, its application to the genus 0 mirror conjecture, a mirror theorem for concave bundle spaces over toric manifolds generalizing a recent result of B. Lian, K. Liu and S.-T. Yau. We also establish a correspondence (see the extensive footnote in section 4) between their new proof of the genus 0 mirror conjecture for quintic 3-folds and our proof of the same conjecture given two years ago. | Elliptic Gromov - Witten invariants and the generalized mirror
conjecture | 11,739 |
In this note, we would like to propose a suitable extension of the arithmetic Chow group of codimension one, in which the Hodge index theorem holds. We also prove an arithmetic analogue of Bogomolov's instability theorem for rank 2 vector bundles on arbitrary regular projective arithmetic varieties. | Intersection pairing for arithmetic cycles with degenerate Green
currents | 11,740 |
We describe a relation between Arnold's strange duality and a polar duality between the Newton polytopes which is mostly due to M.~Kobayashi. We show that this relation continues to hold for the extension of Arnold's strange duality found by C.~T.~C.~Wall and the author. By a method of Ehlers-Varchenko, the characteristic polynomial of the monodromy of a hypersurface singularity can be computed from the Newton diagram. We generalize this method to the isolated complete intersection singularities embraced in the extended duality. We use this to explain the duality of characteristic polynomials of the monodromy discovered by K.~Saito for Arnold's original strange duality and extended by the author to the other cases. | Strange duality and polar duality | 11,741 |
The aim of this paper is to find all algebraic threefolds admitting quasi-regular Poisson structure. There are three types of such varieties: abelian varieties, smooth flat conic bundles over abelian surfaces and quotients of the product of a curve and a symplectic surface by a finite group. | Poisson structures on algebraic threefolds | 11,742 |
The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a "cohomology with coefficients in representations" and a Chern character, and we prove a Grothendieck-Riemann-Roch theorem for the Riemann-Roch transformation it defines. As a corollary we obtain an Hirzebruch-Riemann-Roch formula for the Euler characteristic of a coherent sheaf, and some formulas for the different topological Euler characteristics of complex algebraic stacks. | Riemann-Roch Theorems for Deligne-Mumford Stacks | 11,743 |
We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on surfaces within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilbert schemes of the affine plane with a ring of differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilb^n up to n=7, and give a conjecture for the generating series. | Chern Classes of Tautological Sheaves on Hilbert Schemes | 11,744 |
In this paper we prove that the Gorenstein cyclic quotient singularities of type \frac 1l (1,..., 1,l-(r-1)) with $l\geq r\geq 2$, have a \textit{unique}torus-equivariant projective, crepant, partial resolution, which is ``full'' iff either $l\equiv 0$ mod $% (r-1) $ or $l\equiv 1$ mod $(r-1) $. As it turns out, if one of these two conditions is fulfilled, then the exceptional locus of the full desingularization consists of $\lfloor \frac{l}{r-1} \rfloor$ prime divisors, $\lfloor \frac{l}{r-1}\rfloor - 1$ of which are isomorphic to the total spaces of $\Bbb{P}_{\Bbb{C}}^1$-bundles over $\Bbb{P}_{\Bbb{C}%}^{r-2}$. Moreover, it is shown that intersection numbers are computable explicitly and that the resolution morphism can be viewed as a composite of successive (normalized) blow-ups. Obviously, the monoparametrized singularity-series of the above type contains (as its ``first member'') the well-known Gorenstein singularity defined by the origin of the affine cone which lies over the $r$-tuple Veronese embedding of $\Bbb{P}_{\Bbb{C}}^{r-1}$. | On a series of Gorenstein cyclic quotient singularities admitting a
unique projective crepant resolution | 11,745 |
An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces $\Bbb{C}^{r}/G$ in dimensions $r\geq 4$ would primarily demand the existence of projective, crepant, full desingularizations. Since this is not always possible, it is natural to ask about special classes of such quotient spaces which would satisfy the above property. In this paper we give explicit necessary and sufficient conditions under which 2-parameter series of Gorenstein cyclic quotient singularities have torus-equivariant resolutions of this specific sort in all dimensions. | On crepant resolutions of 2-parameter series of Gorenstein cyclic
quotient singularities | 11,746 |
Given a generic field extension F/k of degree n>3 (i.e. the Galois group of the normal closure of F is isomorphic to the symmetric group $S_n$), we prove that the norm torus, defined as the kernel of the norm map $N:R_{F/k}(G_m)\to\G_m$, is not rational over k. | On a conjecture of Le Bruyn | 11,747 |
In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds $F(n_1, ..., n_l, n)$. This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of $F(n_1, ..., n_l, n)$ to a certain Gorenstein toric Fano variety $P(n_1, ..., n_l, n)$ which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of $P(n_1, ..., n_l, n)$ and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of $P(n_1, ..., n_l, n)$. | Mirror Symmetry and Toric Degenerations of Partial Flag Manifolds | 11,748 |
Let $A$ be a noetherian ring and $R_A$ be the graded ring $A[X,Y,Z,T]$. In this article we introduce the notion of a triad, which is a generalization to families of curves in ${\bf P}^3_A$ of the notion of Rao module. A triad is a complex of graded $R_A$-modules $(L_1 \to L_0 \to L_{-1})$ with certain finiteness hypotheses on its cohomology modules. A pseudo-isomorphism between two triads is a morphism of complexes which induces an isomorphism on the functors $ h_0 (L\otimes .)$ and a monomorphism on the functors $h_{-1} (L\otimes .)$. One says that two triads are pseudo-isomorphic if they are connected by a chain of pseudo-isomorphisms. We show that to each family of curves is associated a triad, unique up to pseudo-isomorphism, and we show that the map $\{\hbox{families of curves}\}\to \{\hbox{triads}\}$ has almost all the good properties of the map $\{\hbox{curves}\}\to \{\hbox{Rao modules}\}$. In a section of examples, we show how to construct triads and families of curves systematically starting from a graded module and a sub-quotient (that is a submodule of a quotient module), and we apply these results to show the connectedness of $H_{4,0}$. | Triades et familles de courbes gauches | 11,749 |
We define, in a purely algebraic way, 1-motives $Alb^{+}(X)$, $Alb^{-}(X)$, $Pic^{+}(X)$ and $Pic^{-}(X)$ associated with any algebraic scheme $X$ over an algebraically closed field of characteristic zero. For $X$ over $\C$ of dimension $n$ the Hodge realizations are, respectively, $H^{2n-1}(X)(n)$, $H_{1}(X)$, $H^{1}(X)(1)$ and $H_{2n-1}(X)(1-n)$. | Albanese and Picard 1-motives | 11,750 |
We show how Chern classes of a Calabi Yau hypersurface in a toric Fano manifold can be expressed in terms of the holomorphic at a maximal degeneracy point period of its mirror. We also consider the relation between Chern classes and the periods of mirrors for complete intersections in Grassmanian Gr(2,5). | Chern classes and the periods of mirrors | 11,751 |
Let $G$ be a semisimple complex Lie group with a Borel subgroup $B$. Let $X=G/B$ be the flag manifold of $G$. Let $C=P^1\ni\infty$ be the projective line. Let $\alpha\in H_2(X,{\Bbb Z})$. The moduli space of $G$-monopoles of topological charge $\alpha$ (see e.g. [Jarvis]) is naturally identified with the space $M_b(X,\alpha)$ of based maps from $(C,\infty)$ to $(X,B)$ of degree $\alpha$. The moduli space of $G$-monopoles carries a natural hyperk\"ahler structure, and hence a holomorphic symplectic structure. We propose a simple explicit formula for the symplectic structure on $M_b(X,\alpha)$. It generalizes the well known formula for $G=SL_2$ (see e.g. [Atiyah-Hitchin]). Let $P\supset B$ be a parabolic subgroup. The construction of the Poisson structure on $M_b(X,\alpha)$ generalizes verbatim to the space of based maps $M=M_b(G/P,\beta)$. In most cases the corresponding map $T^*M\to TM$ is not an isomorphism, i.e. $M$ splits into nontrivial symplectic leaves. These leaves are explicilty described. | A note on the symplectic structure on the space of G-monopoles | 11,752 |
The recent two proofs for the (weak) factorization theorem for birational maps, one by W{\l}odarczyk and the other by Abramovich-Karu-Matsuki-W{\l}odarczyk rely on the results of Morelli. The former uses the process for $\pi$-desingularization (the most subtle part of Morelli's combinatorial algorithm), while the latter uses the strong factorization of toric (toroidal) birational maps directly. This paper provides a coherent account of Morelli's work (and its toroidal extension) clarifying some discrepancies in the original argument. | A note on the factorization theorem of toric birational maps after
Morelli and its toroidal extension | 11,753 |
Chisini's conjecture asserts that for a cuspidal curve $B\subset \mathbb P^2$ a generic morphism $f$ of a smooth projective surface onto $\mathbb P^2$ of degree $\geq 5$, branched along $B$, is unique up to isomorphism. We prove that if $\deg f$ is greater than the value of some function depending on the degree, genus, and number of cusps of $B$, then the Chisini conjecture holds for $B$. This inequality holds for many different generic morphisms. In particular, it holds for a generic morphism given by a linear subsystem of the $m$th canonical class for almost all surfaces with ample canonical class. | On a Chisini Conjecture | 11,754 |
Here we prove that the minimal free resolution of a general space curve of large degree (e.g. a general space curve of degree d and genus g with d g+3, except for finitely many pairs (d,g)) is the expected one. A similar result holds even for general curves with special hyperplane section and, roughly, d g/2. The proof uses the so-called methode d'Horace. | On the minimal free resolution of non-special curves in P^3 | 11,755 |
Given a projective algebraic variety $X$, let $\Pi_p(X)$ denote the monoid of effective algebraic equivalence classes of effective algebraic cycles on $X$. The $p$-th Euler-Chow series of $X$ is an element in the formal monoid-ring $Z[[\Pi_p(X)]]$ defined in terms of Euler characteristics of the Chow varieties $\cvpd{p}{\alpha}{X}$ of $X$, with $\alpha \in\Pi_p(X)$. We provide a systematic treatment of such series, and give projective bundle formulas which generalize previous results by B. Lawson and S.S.Yau and Elizondo. The techniques used involve the Chow quotients introduced by Kapranov, and this allows the computation of various examples including some Grassmannians and flag varieties. There are relations between these examples and representation theory, and further results point to interesting connections between Euler-Chow series for certain varieties and the topology of the closure of moduli spaces $M_{0,n+1}$. | Chow quotients and projective bundle formulas for Euler-Chow series | 11,756 |
We give a geometric interpretation of the base change homomorphism between the Hecke algebra of GL(n) for an unramified extension of local fields of positive characteristic. For this, we use some results of Ginzburg, Mirkovic and Vilonen related to the geometric Satake isomorphism. We give new proof for these results in the positive characteristic case. By using that geometric interpretation of the base change homomorphism, we prove the fundamental lemma of Jacquet and Ye for arbitrary Hecke function in the the equal characteristic case. | Faisceaux pervers, homomorphisme de changement de base et lemme
fondamental de Jacquet et Ye | 11,757 |
In this article we address the problem of computing the dimension of the space of plane curves of degree $d$ with $n$ general points of multiplicity $m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple $(-1)$-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed in a previous article ("Degenerations of Planar Linear Systems", alg-geom/9702015) to show that the conjecture holds for all $m \leq 12$. | Linear Systems of Plane Curves with Base Points of Equal Multiplicity | 11,758 |
We study the vector bundles without intermediate cohomology on Fano threefolds of index two, degree d=3,4,5 and Betti number one. We obtain a complete characterization in the case of rank-two vector bundles. For arbitrary rank, we give all possible Chern classes of such vector bundles, under some general condition. | Vector bundles on Fano 3-folds without intermediate cohomology | 11,759 |
In this paper a formula is proved for the general degeneracy locus associated to an oriented quiver of type A_n. Given a finite sequence of vector bundles with maps between them, these loci are described by putting rank conditions on arbitrary composites of the maps. Our answer is a polynomial in Chern classes of the bundles involved, depending on the given rank conditions. It can be expressed as a linear combination of products of Schur polynomials in the differences of the bundles. The coefficients are interesting generalizations of Littlewood-Richardson numbers. These polynomials specialize to give new formulas for Schubert polynomials. | Chern class formulas for quiver varieties | 11,760 |
In the first part of the paper, we give an explicit algorithm to compute the (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary convex projective variety in some points if one knows the Gromov-Witten invariants of the original variety. In the second part, we specialize to blow-ups of P^r and show that many invariants of these blow-ups can be interpreted as numbers of rational curves on P^r having specified global multiplicities or tangent directions in the blown-up points. We give various numerical examples, including a new easy way to determine the famous multiplicity d^{-3} for d-fold coverings of rational curves on the quintic threefold, and, as an outlook, two examples of blow-ups along subvarieties, whose Gromov-Witten invariants lead to classical multisecant formulas. | Gromov-Witten invariants of blow-ups | 11,761 |
We consider a class of stable smoothable n-dimensional varieties, the analogs of stable curves. Assuming the minimal model program in dimension n+1, we prove that this class is bounded. From Kollar's method of constructing projective moduli spaces we get as a corollary that minimal model program in dimension n+1 implies the existence of a projective coarse moduli space for stable smoothable n-folds. | Minimal models and boundedness of stable varieties | 11,762 |
We show that birational smooth complex projective varieties with numerically effective canonical bundles along the exceptional loci have the same Betti numbers. In particular, birational smooth minimal models share the same Betti numbers. Our methods indicate a link between $p$-adic measure and the minimal model theory. | On the Topology of Birational Minimal Models | 11,763 |
A meromorphic function on a compact complex analytic manifold defines a $\bc\infty$ locally trivial fibration over the complement of a finite set in the projective line $\bc\bp^1$. We describe zeta-functions of local monodromies of this fibration around atypical values. Some applications to polynomial functions on $\bc^n$ are described. | On atypical values and local monodromies of meromorphic functions | 11,764 |
Let X be a smooth projective surface. Here we study the postulation of a general union Z of fat points of X, when most of the connected components of Z have multiplicity 2. This problem is related to the existence of "good" families of curves on X, with prescribed singularities, most of them being nodes, and to the cohomology of suitable line bundles on blowing ups of X. More precise statements are obtained in the case X=P^2. | Nodal curves and postulation of generic fat points on surfaces | 11,765 |
Families of stable curves of genus $\gamma$ over a smooth curve $C$ correspond to morphisms from $C$ to the moduli stack of stable curves $\bar{\cal M}_\gamma$. It is natural to compactify the corresponding moduli problem using stable maps into the stack. In order to get a complete moduli problem, the source curves must acquire extra structure, and you will have to read the paper to find what that is. In this paper, we define these stable maps into $\bar{\cal M}_\gamma$ in characteristic 0, and show that they form a proper Deligne-Mumford stack admitting a projective coarse moduli space. A comparison with Alexeev's work is given. Natural generalizations for stable maps into other Deligne-Mumford stacks will appear in a follow up paper, along with applications to areas such as admissible covers, level structures and higher dimensional families. | Complete moduli for fibered surfaces | 11,766 |
We investigate the possibility of embedding minimal abelian surfaces in smooth toric 4-folds with Picard number 2. The existence of such an embedding imposes conditions on the 4-fold, which we partly describe. On the other hand, we exhibit such embeddings in a special case, using a simple description of morphisms into smooth toric varieties. | Abelian surfaces in toric 4-folds | 11,767 |
Castelnuovo-Mumford regularity is an important invariant of projective algebraic varieties. A well known conjecture due to Eisenbud and Goto gives a bound for regularity in terms of the codimension and degree,i.e., Castelnuovo-Mumford regularity of a given variety X is less than or equal to deg(X)-codimension(X)+1. Generic projection methods proved to be effective for the study of regularity of smooth projevtive varieties of dimension at most four(cf.[BM},[K2],[L],[Pi] and [R1]) because there are nice vanishing theorems for cohomology of vector bundles (e.g. the Kodaira-Kawamata-Viehweg vanishing theorem) and detailed information about the fibers ofgeneric projections from X to a hypersurface of the same dimension. Here we show by using methods similar to those used in [K2] that $\reg{X}\le(deg(X)-codimension(X)+1)+10$ for any smooth fivefold and $\reg{X}\le(deg(X)-codimension(X)+1)+20$ for any smooth sixfold. Furthermore, using similar methods we give a bound for the regularity of an arbitrary (not necessarily locally Cohen-Macaulay) projective surface X in P^N. To wit, we show that $\reg{X}\le(d-e+1)d-(2e+1)$, where d=deg(X) and e=codimension(X). This is the first bound for surfaces which does not depend on smoothness. | Generic Projection Methods in Castelnuovo Regularity of Projective
Varieties | 11,768 |
Let $f: X \to S$ be flat morphism over an algebraically closed field $k$ with a relative normal crossings divisor $Y\subset X$, $(E, \nabla)$ be a bundle with a connection with log poles along $Y$ and curvature with values in $f^*\Omega^2_{k(S)}$. Then the Gau\ss-Manin sheaf $R^if_*(\Omega^*_{X/S}({\rm log} Y)\otimes E)$ carries a Gau\ss-Manin connection $GM^i(\nabla)$. We establish a Riemann-Roch formula relating the algebraic Chern-Simons invariants of $\nabla$, $GM^i(\nabla)$ and the top Chern class of $\Omega^1_{X/S}({\rm log}Y)$. | A Riemann-Roch theorem for flat bundles, with values in the algebraic
Chern-Simons theory | 11,769 |
For a quasiprojective variety S, we define a category CHM(S) of pure Chow motives over S. Assuming conjectures of Grothendieck and Murre, we show that the decomposition theorem holds in CHM(S). As a consequence, the intersection complex of S makes sense as an object IS of CHM(S). In the forthcoming part II we will give an unconditional definition of intersection Chow groups and study some of their properties. | Motivic Decomposition and Intersection Chow Groups I | 11,770 |
In 1988 S. Mori, D. Morrison, and I. Morrison gave a computer-based conjectural classification of four-dimensional cyclic quotient singularities of prime index. It was partially proven in 1990 by G. Sankaran. In 1991 Jim Lawrence basically proved this and much more, been unaware of algebro-geometric meaning of what he did. In this short note I just bring this all together to prove that all n-dimensional toric singularities with log-discrepancy greater than (or equal to) $\epsilon$ form finitely many "series". | On classification of toric singularities | 11,771 |
We construct the canonical structure of an irreducible projective variety on the set of connected curves of degree $d$ in $\Bbb P^n$ with rational components (some components can be multiple). The set of rational curves is open subset in the Zariski topology under this structure. | On the Variety of Rational Curves of Degree $d$ in $\Bbb P^n$ | 11,772 |
We make a detailed analysis of the A-hypergeometric system (or GKZ system) associated with a monomial curve and integral, hence resonant, exponents. We characterize the Laurent polynomial solutions and show that these are the only rational solutions. We also show that for any exponent, there are at most two linearly independent Laurent solutions, and that the upper bound is reached if and only if the curve is not arithmetically Cohen--Macaulay. We then construct, for all integral parameters, a basis of local solutions in terms of the roots of the generic univariate polynomial associated with A. We determine the holonomic rank r for all integral exponents and show that it is constantly equal to the degree d of X if and only if X is arithmetically Cohen-Macaulay. Otherwise there is at least one exponent for which r = d + 1. | The A-hypergeometric System Associated with a Monomial Curve | 11,773 |
We describe algorithms for computing various functors for algebraic D-modules, i.e. systems of linear partial differential equations with polynomial coefficients. We will give algorithms for restriction, tensor product, localization, and local cohomology groups for all degrees. | Algorithms for D-modules --- restriction, tensor product, localization,
and local cohomology groups | 11,774 |
The `linear orbit' of a plane curve of degree d is its orbit in the projective space of dimension d(d+3)/2 parametrizing such curves under the natural action of PGL(3). In this paper we compute the degree of the closure of the linear orbits of most curves with positive dimensional stabilizers. Our tool is a nonsingular variety dominating the orbit closure, which we construct by a blow-up sequence mirroring the sequence yielding an embedded resolution of the curve. The results given here will serve as an ingredient in the computation of the analogous information for arbitrary plane curves. Linear orbits of smooth plane curves are studied in [A-F1]. | Plane curves with small linear orbits I | 11,775 |
We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on $Pic^{g-1}C$ which are linearly equivalent to $2\Theta$. The embedded tangent space at a semi-stable non-stable bundle $\xi\oplus\xi^{-1}$, where $\xi$ is a degree zero line bundle, is shown to consist of those divisors in $|2\Theta|$ which contain $Sing(\Theta_{\xi})$ where $\Theta_{\xi}$ is the translate of $\Theta$ by $\xi$. We also obtain geometrical results on the structure of this tangent space. | The tangent space to the moduli space of vector bundles on a curve and
the singular locus of the theta divisor of the jacobian | 11,776 |
Let X be a projective complex 3-fold, quasihomogeneous with respect to an action of a linear algebraic group. We show that X is a compactification of SL_2/G, G a discrete subgroup, or that X can be equivariantly transformed into the 3-dim. projective space, the quadric Q_3, or into certain quasihomogeneous bundles with very simple structure. | Simple Models of Quasihomogeneous Projective 3-Folds | 11,777 |
In the present work we classify the relatively minimal 3-dimensional quasihomogeneous complex projective varieties under the assumption that the automorphism group is not solvable. By relatively minimal we understand varieties X having at most Q-factorial terminal singularities and allowing an extremal contraction X to Y, where dim Y <3. | Relatively Minimal Quasihomogeneous Projective 3-Folds | 11,778 |
In the present work we describe 3-dimensional complex SL_2-varieties where the generic SL_2-orbit is a surface. We apply this result to classify the minimal 3-dimensional projective varieties with Picard-number 1 where a semisimple group acts such that the generic orbits are 2-dimensional. This is an ingredient of the classification [Keb98, math.AG/9805042] of the 3-dimensional relatively minimal quasihomogeneous varieties where the automorphism group is not solvable. | On the Classification of 3-dimensional SL_2(C)-varieties | 11,779 |
This is the first part of two papers. In this part, we prove the blowup formulae for virtual Hodge polynomials of Gieseker moduli spaces of rank-2 stable sheaves and Uhlenbeck compactification spaces over algebraic surfaces. In particular, we verify the blowup formulae for the S-duality conjecture of Vafa-Witten, i.e. the blowup formulae for the Euler numbers of instanton moduli spaces over algebraic surfaces. | On blowup formulae for the S-duality conjecture of Vafa and Witten | 11,780 |
This is the second part of two papers. In this part, we establish closed formulae for the universal functions in the blowup formulae for virtual Hodge polynomials of Gieseker moduli spaces of rank-2 stable sheaves and Uhlenbeck compactification spaces over algebraic surfaces. | On blowup formulae for the S-duality conjecture of Vafa and Witten II:
the universal functions | 11,781 |
We extend the methods of geometric invariant theory to actions of non--reductive groups in the case of homomorphisms between decomposable sheaves whose automorphism groups are non--reductive. Given a linearization of the natural action of the group $\Aut(E)\times\Aut(F)$ on Hom(E,F), a homomorphism is called stable if its orbit with respect to the unipotent radical is contained in the stable locus with respect to the natural reductive subgroup of the automorphism group. We encounter effective numerical conditions for a linearization such that the corresponding open set of semi-stable homomorphisms admits a good and projective quotient in the sense of geometric invariant theory, and that this quotient is in addition a geometric quotient on the set of stable homomorphisms. | Moduli spaces of decomposable morphisms of sheaves and quotients by
non-reductive groups | 11,782 |
This is a report on some recent work by Gaffney, Massey, and the author, characterizing the conditions A_f and W_f for a family of ICIS germs equipped with a function. First we introduce the work informally. Then we review the formal definitions of A_f and W_f, and state the theorems that characterize them by the constancy of Milnor numbers. Next we review the definition of the Buchsbaum-Rim multiplicity, and reformulate the theorems by the constancy of certain Buchsbaum-Rim multiplicities. Finally, we review the theory of integral dependence of elements on submodules of free modules, and apply it to prove the reformulated theorems. | Equisingularity, Multiplicity, and Dependence | 11,783 |
We classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group. We also classify the orbits in each case and construct explicit representatives. This generalizes the classical Schubert decompostion, which states that the GL(n)-orbits on a product of two flag varieties correspond to permutations. Our main tool is the theory of quiver representations. | Multiple Flag Varieties of Finite Type | 11,784 |
We consider an equisingularity problem for polynomial families of affine hypersurfaces $X_\tau \subset \mathbb C^n$ with (at worst) isolated singularities. We show that the constancy of the global polar invariants $\gamma^* (X_\tau)$ is equivalent to the $t$-equisingularity at infinity, an asymptotic-type equisingularity that we introduce. We prove that $\gamma^*$-constancy implies C$^\infty$-triviality in the neighbourhood of infinity. We show how the invariants $\gamma^*$ enter in the description of a CW-complex model of a hypersurface $X_\tau$ and therefore provide in particular new invariants at infinity for polynomial functions $f: \mathbb C^n \to \mathbb C$. | Asymptotic equisingularity and topology of complex hypersurfaces | 11,785 |
We analyse the Gauss-Manin system of differential equations---and its Fourier transform---attached to regular functions satisfying a tameness assupmption on a smooth affine variety over C (e.g. tame polynomials on C^{n+1}). We give a solution to the Birkhoff problem and prove Hodge-type results analogous to those existing for germs of isolated hypersurface singularities. | Hypergeometric periods for a tame polynomial | 11,786 |
By a K3-surface with nine cusps I mean a compact complex surface with nine isolated double points $A_2$, but otherwise smooth, such that its minimal desingularisation is a K3-surface. In an earlier paper I showd that each such surface is a quotient of a complex torus by a cyclic group of order three. Here I try to classify these $K3$-surfaces, using the period map for complex tori. In particular I show: A $K3$-surface with nine cusps carries polarizations only of degrees 0 or 2 modulo 6. This implies in particular that there is no quartic surface in projective three-space with nine cusps. (T. Urabe pointed out to me how to deduce this from a theorem of Nikulin.) In an appendix I give explicit equations of quartic surfaces in three-space with eight cusps. | On the Classification of K3-Surfaces with Nine Cusps | 11,787 |
We show that, given a projective regular function f on a smooth quasi-projective variety over C, the corresponding cohomology groups of the algebraic de Rham complex with twisted differential d-df and of the complex of algebraic forms with differential df have the same dimension (a result announced by Barannikov and Kontsevitch). We generalize the result to de Rham complexes with coefficients in a mixed Hodge Module. | On a twisted de Rham complex | 11,788 |
Let $f_i$ be polynomials in $n$ variables without a common zero. Hilbert's Nullstellensatz says that there are polynomials $g_i$ such that $\sum g_if_i=1$. The effective versions of this result bound the degrees of the $g_i$ in terms of the degrees of the $f_j$. The aim of this paper is to generalize this to the case when the $f_i$ are replaced by arbitrary ideals. Applications to the B\'ezout theorem, to \L ojasiewicz--type inequialities and to deformation theory are also discussed. | Effective Nullstellensatz for Arbitrary Ideals | 11,789 |
It has been proved several times in the literature that a polynomial map from $C^2$ to $C$ with irreducible rational fibers cannot be a component of a counterexample to the Jacobian Conjecture. This note points out that this result is empty: it is implicit in 1980 work of Miyanish and Sugie that such a polynomial is equivalent to $f(x,y)=x$ by a polynomial automorphism of $C^2$. | Nontrivial rational polynomials in two variables have reducible fibres | 11,790 |
These are the informal notes of two seminars held at the Universita` di Roma "La Sapienza", and at the Scuola Normale Superiore in Pisa in Spring and Autumn 1997. We discuss in detail the content of the parts of Givental's paper dealing with mirror symmetry for projective complete intersections. | On the work of Givental relative to mirror symmetry | 11,791 |
In preprint alg-geom/9708009 we have constructed a (ten-dimensional) symplectic desingularization of the moduli space of rank-two torsion-free semistable sheaves on a $K3$, with trivial determinant and second Chern class equal to 4. In the present paper we prove that (every deformation of) this desingularization is a new irreducible symplectic variety. More precisely we show that the desingularization is irreducible symplectic, and that its second Betti number is at least 24. | Desingularized moduli spaces of sheaves on a K3, II | 11,792 |
The purpose of these notes is to give an introduction to Deligne-Mumford stacks and their moduli spaces, with emphasis on the moduli problem for curves. The paper has 4 sections. In section 1 we discuss the general problem of constructing a moduli "space" of curves. In section 2 we give an introduction to Deligne-Mumford stacks and their moduli spaces. In section 3 we return to curves and outline Deligne and Mumford's proof that the stack of stable curves is a smooth and irreducible Deligne-Mumford stack which is proper over Spec Z. In section 4 we explain how to use geometric invariant to construct moduli spaces for quotient stacks. Finally, we briefly outline Gieseker's geometric invariant theory construction of the moduli scheme of projective curves defined over an algebraically closed field. These notes are a slightly revised version of notes which the author has circulated privately for several years, and are based on lectures the author gave at the Weizmann Institute in July 1994. They are also available at the URL http://math.missouri.edu/~edidin/Papers/ Any updates will be posted to this URL. | Notes on the construction of the moduli space of curves | 11,793 |
One of the simplest examples of a smooth, non degenerate surface in P^4 is the quintic elliptic scroll. It can be constructed from an elliptic normal curve E by joining every point on E with the translation of this point by a non-zero 2-torsion point. The same construction can be applied when E is replaced by a (lineaerly normally embedded) abelian variety A. In this paper we ask the question when the resulting scroll Y is smooth. If A is an abelian surface embedded by a line bundle L of type (d_1,d_2) and r=d_1d_2, then we prove that for general A the scroll Y is smooth if r is at least 7 with the one exception where r=8 and the 2-torsion point is in the kernel K(L) of L. In this case Y is singular.The case r=7 is particularly interesting, since then Y is a smooth threefold in P^6 with irregularity 2. The existence of this variety seems not to have been noticed before. One can also show that the case of the quintic elliptic scroll and the above case are the only possibilities where Y is smooth and the codimension of Y is at most half the dimension of the surrounding projective space. | A Series of Smooth Irregular Varieties in Projective Space | 11,794 |
We define a quasi--projective reduction of a complex algebraic variety $X$ to be a regular map from $X$ to a quasi--projective variety that is universal with respect to regular maps from $X$ to quasi--projective varieties. A toric quasi--projective reduction is the analogous notion in the category of toric varieties. For a given toric variety $X$ we first construct a toric quasi--projective reduction. Then we show that $X$ has a quasi--projective reduction if and only if its toric quasi--projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi--projective toric variety admits a categorical quotient in the category of quasi--projective varieties. | Quasi--Projective Reduction of Toric Varieties | 11,795 |
We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over the complex numbers all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When we say that the curves can be constructed ``explicitly'', we mean that the coefficients of the defining equations of the curves are simple rational expressions in algebraic numbers in R whose minimal polynomials over Q can be given exactly and whose decimal approximations can be given to as many places as is necessary to distinguish them from their conjugates. We also prove a simply-stated theorem that allows one to decide whether or not two plane quartics over the complex numbers, each with a pair of commuting involutions, are isomorphic to one another. | Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian | 11,796 |
In this paper we give for all $n \geq 2$, d>0, $g \geq 0$ necessary and sufficient conditions for the existence of a pair (X,C), where X is a K3 surface of degree 2n in $\matbf{P}^{n+1}$ and C is a smooth (reduced and irreducible) curve of degree d and genus g on X. The surfaces constructed have Picard group of minimal rank possible (being either 1 or 2), and in each case we specify a set of generators. For $n \geq 4$ we also determine when X can be chosen to be an intersection of quadrics (in all other cases X has to be an intersection of both quadrics and cubics). Finally, we give necessary and sufficient conditions for $\O_C (k)$ to be non-special, for any integer $k \geq 1$. | Smooth curves on projective K3 surfaces | 11,797 |
The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi--Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries are parallel to classical gauge theories and their complexifications. This article explains this parallelism, so that the appearance of invariants of new type in complexified gauge theory (see Donaldson--Thomas [D-T] and Thomas [T]) can be accompanied by analogous invariants in the theory of special Lagrangian cycles, for which the development is at present much more modest than in gauge theory. We discuss related geometric constructions, arising from mirror symmetry and symplectic geometry. | Special Langrangian geometry and slightly deformed algebraic geometry
(spLag and sdAG) | 11,798 |
Barth and Nieto have found a remarkable quintic threefold which parametrizes Heisenberg invariant Kummer surfaces which belong to abelian surfaces with a (1,3)-polarization and a lecel 2 structure. A double cover of this quintic, which is also a Calabi-Yau variety, is birationally equivalent to the moduli space {\cal A}_3(2) of abelian surfaces with a (1,3)-polarization and a level 2 structure. As a consequence the corresponding paramodular group \Gamma_3(2) has a unique cusp form of weight 3. In this paper we find this cusp form which is \Delta_1^3. The form \Delta_1 is a remarkable weight 1 cusp form with a character with respect to the paramodular group \Gamma_3. It has several interesting properties. One is that it admits an infinite product representation, the other is that it vanishes of order 1 along the diagonal in Siegel space. In fact \Delta_1 is an element of a short series of modular forms with this last property. Using the fact that \Delta_1 is a weight 3 cusp form with respect to the group \Gamma_3(2) we give an independent construction of a smooth projective Calabi-Yau model of the moduli space {\cal A}_3(2). | The Modular Form of the Barth-Nieto Quintic | 11,799 |
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