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The article starts with some introductory material about resolution graphs of normal surface singularities (definitions, topological/homological properties, etc). We then discuss the case when the normal surface singularity is an N-fold cyclic covering of a surface germ, branched along a curve given by the germ of an analytic function f. We present non-trivial examples in order to show that from the embedded resolution graph G of f in general it is not possible to recover the resolution graph of the cyclic covering. The main results are the construction of a ``universal covering graph'' from the topology of the germ f, and the completely combinatorial construction of the resolution graph of cyclic coverings from this universal graph of f and the integer N. For this we also prove some purely graph-theoretical classification theorems of ``covering graphs''. In the last part, we connect the properties of the universal covering graph with the topological invariants of f, e.g. with the nilpotent part of its algebraic monodromy. | Resolution graphs of some surface singularities I. (cyclic coverings) | 12,200 |
In this article we give a construction of the resolution graphs of hypersurface surface singularities (X_k,0) given by generalized Iomdin series. All these resolution graphs are coordinated by an ``universal bi-colored graph'' which is associated with the ICIS determining the Iomdin series. The definition of this new graph is rather involved, and in concrete examples it is difficult to compute. Nevertheless, we present a large number of examples. This is very helpful in the exemplification of its properties as well. We then present a construction of the resolution graphs of the surface singularities above which uses the "universal bi-coloured graph" and the integer k. This is formulated in a purely combinatorial algorithm. The result is a highly non-trivial generalization of the case of cyclic coverings of smooth surfaces. | Resolution graphs of some surface singularities II. (generalized Iomdin
series) | 12,201 |
In this paper we prove the Tamagawa number conjecture of Bloch and Kato for CM elliptic curves using a new explicit description of the specialization of the elliptic polylogarithm. The Tamagawa number conjecture describes the special values of the L-function of a CM elliptic curve in terms of the regulator maps of the K-theory of the variety into Deligne and etale cohomology. The regulator map to Deligne cohomology was computed by Deninger with the help of the Eisenstein symbol. For the Tamagawa number conjecture one needs an understanding of the $p$-adic regulator on the subspace of K-theory defined by the Eisenstein symbol. This is accomplished by giving a new explicit computation of the specialization of the elliptic polylogarithm sheaf. It turns out that this sheaf is an inverse limit of $p^r$-torsion points of a certain one-motive. The cohomology classes of the elliptic polylogarithm sheaf can then be described by classes of sections of certain line bundles. These sections are elliptic units and going carefully through the construction one finds an analog of the elliptic Soul\'e elements. Finally Rubin's ``main conjecture'' of Iwasawa theory is used to compare these elements with etale cohomology. | The Tamagawa number conjecture for CM elliptic curves | 12,202 |
Let $X$ be a smooth hypersurface of degree $n\geq 3$ in $\mathbb{P}^n$. We prove that the log canonical threshold of $H\in|-K_X|$ is at least $\frac{n-1}{n}$. Under the assumption of the Log minimal model program, we also prove that a hyperplane section $H$ of $X$ is a cone in $\mathbb{P}^{n-1}$ over a smooth hypersurface of degree $n$ in $\mathbb{P}^{n-2}$ if and only if the log canonical threshold of $H$ is $\frac{n-1}{n}$. | Log Canonical Thresholds and Generalized Eckardt Points | 12,203 |
Let U be an open subset of a unirational variety. We prove that there is rational curve C in U such that the fundamental group of C surjects onto the fundamental group of U. As a consequence we obtain new proofs of the theorems of Harbater and Colliot-Th\'el\`ene on Galois covers and torsors over the p-adic line. We also obtain examples of pencils of curves of genus <14 whose monodromy group is the full Teichm\"uller group. | Fundamental groups of rationally connected varieties | 12,204 |
Let us consider the rank 14 lattice $P=D_4^3\oplus < -2> \oplus < 2>$. We define a K3 surface S of type P with the property that $P\subset {\rm Pic}(S) $, where ${\rm Pic}(S) $ indicates the Picard lattice of S. In this article we study the family of K3 surfaces of type P with a certain fixed multipolarization. We note the orthogonal complement of P in the K3 lattice takes the form $$ U(2)\oplus U(2)\oplus (-2I_4). $$ We show the following results: \item{(1)} A K3 surface of type P has a representation as a double cover over ${\bf P}^1\times {\bf P}^1$ as the following affine form in (s,t,w) space: $$ S=S(x): w^2=\prod_{k=1}^4 (x_{1}^{(k)}st+x_{2}^{(k)}s+x_{3}^{(k)}t+x_{4}^{(k)}), \ x_k=\pmatrix{x_{1}^{(k)}&x_{2}^{(k)}\cr x_{3}^{(k)}&x_{4}^{(k)}} \in M(2,{\bf C}). $$ We make explicit description of the Picard lattice and the transcendental lattice of S(x). \item{(2)} We describe the period domain for our family of marked K3 surfaces and determine the modular group. \par \noindent \item{(3)} We describe the differential equation for the period integral of S(x) as a function of $x\in (GL(2,{\bf C}))^4$. That bocomes to be a certain kind of hypergeometric one. We determine the rank, the singular locus and the monodromy group for it. \par \noindent \item{(4)} It appears a family of 8 dimensional abelian varieties as the family of Kuga-Satake varieties for our K3 surfaces. The abelian variety is characterized by the property that the endomorphism algebra contains the Hamilton quarternion field over ${\bf Q}$. | Study on the family of K3 surfaces induced from the lattice $(D_4)^3
\oplus < -2 > \oplus < 2 > | 12,205 |
Let R be a Dedekind scheme, $\nu$ its generic point, X and V del Pezzo surfaces of degree 1 over R that are Gorenstein Mori fiber spaces (as 3-folds germs over the ground field). We study birational maps $\phi:X\dasharrow V$ over R which are isomorphisms over the generic point of R. We put down normal forms of such transformations (in suitable coordinates) and give some properties of X and V. In particular, we prove the uniqueness of a smooth model. | Gorenstein models of del Pezzo surfaces of degree 1 over Dedekind
schemes | 12,206 |
In this paper we define an action of the Weyl group on the quiver varieties $M_{m,\grl}(d,v)$ with generic $(m,\grl)$. To do it we describe a set of generators of the projective ring of a quiver variety. We also prove connectness for the smooth quiver variety $M(d,v)$ and normality for $M_0(d,v)$ in the case of a quiver of finite type and $d-v$ a regular weight. | A remark on quiver varieties andweyl groups | 12,207 |
We prove the connectedness of the moduli space of maps (of fixed genus and homology class) to the homogeneous space G/P by degeneration via the maximal torus action. In the genus 0 case, the irreducibility of the moduli of maps is a direct consequence of connectedness. An analysis of a related Bialynicki-Birula stratification of the map space yields a rationality result: the (coarse) moduli space of genus 0 maps to G/P is a rational variety. The rationality argument depends essentially upon rationality results for quotients of SL2 representations proven by Katsylo and Bogomolov. | The connectedness of the moduli space of maps to homogeneous spaces | 12,208 |
Let $X$ be a complete normal variety, $B$ an effective $\mathbb{R}$-divisor on $X$, and $D$ a Cartier divisor on $X$. Assume that the pair $(X, B)$ is log terminal. We consider the problem whether $H^0(X, D) \ne 0$ and obtain some results in lower dimensions. | On effective non-vanishing and base-point-freeness | 12,209 |
Suppose X is a smooth projective 3-fold of general type and |mK_X| is composed of a pencil of surfaces with m>1. This pencil naturally induces a fibration f:X->C onto a smooth curve C after the Stein-factorization, which is the main objects of this article. Based on Koll'ar's earlier works, we improve on it and try to understand the family in terms of discrete birational invariants of the total space as well as those of the general fiber and the base curve. The aim of this note is to build a little bit basic facts. | On canonically derived families of surfaces of general type over curves | 12,210 |
Some new results on plane F_{q^2}-maximal curves are stated and proved. It is known that the degree d of such curves is upper bounded by q+1 and that d=q+1 if and only if the curve is F_{q^2}-isomorphic to the Hermitian. We show that d\le q+1 can be improved to d\le (q+2)/2 apart from the case d=q+1 or q\le 5. This upper bound turns out to be sharp for q odd. We also study the maximality of Hurwitz curves of degree n+1. We show that they are F_{q^2}-maximal if and only if (q+1) divides (n^2-n+1). Such a criterion is extended to a wider family of curves. | Remarks on plane maximal curves | 12,211 |
We show that the set of F_q-rational points of either certain Fermat curves or certain F_q-Frobenius non-classical plane curves is a complete (k,d)-arc in P^2(F_q), where k and d are respectively the number of F_q-rational points and the degree of the underlying curve. | On complete arcs arising from plane curves | 12,212 |
In this paper we prove that Arnold Surfaces of all real algebraic curves of even degree with non-empty real part are standard (Rokhlin's Conjecture). There is an obvious connection with classification of Arnold Surfaces up to isotopy of S^4 and Hilbert's Sixteen Problem on the arrangements of connected real components of curves. First, we consider some M-curves, i.e curves of a prescribed degree having the greatest possible number of connected real components, and prove that Arnold surfaces of these curves are standard. Afterwards, we exhibit a procedure of modification "perestroika" of these M-curves which allows to prove the Rokhlin's Conjecture. | Proof of the Rokhlin's Conjecture on Arnold's surfaces | 12,213 |
In the 80's D. Eisenbud and J. Harris posed the following question: "What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type?" We answer their question for one-dimensional families of smooth curves degenerating to stable curves with just two components meeting at points in general position. In this note we treat only those families whose total space is regular. Nevertheless, we announce here our most general answer, to be presented in detail in a forthcoming submission. | Limits of Weierstrass points in regular smoothings of curves with two
components | 12,214 |
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. Now this construction is called the Krichever map. In e-print math.AG/9911097 A.N. Parshin suggested a generalization of the Krichever map for the case of algebraic surfaces from the point of view of 2-dimensional local fields. In this work we suggest a generalization of the Krichever map to the case of algebraic varieties of arbitrary dimension from the point of view of multidimensional local fields. For surfaces our construction coincides with the Parshin construction. Besides, we obtain new explicit acyclic resolutions of quasicoherent sheaves connected with multidimensional local fields. | Krichever correspondence for algebraic varieties | 12,215 |
We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian $G=G(2,N)$ parametrizing lines in projective space, for the natural arithmetic Lefschetz operator defined via the Pl\"ucker embedding of $G$ in projective space. The analysis of the Hodge index inequalities involves estimates on certain Racah polynomials. | Standard conjectures for the arithmetic Grassmannian G(2,N) and Racah
polynomials | 12,216 |
In this article we prove the irreducibility of the Hilbert scheme of rationnal curves on homogeneous varieties with fixed class in the Chow ring. This result has also been proved by J. F. Thomsen [T] and B. Kim and R. Pandharipande [KP]. Our method is totaly different (we don't use the compactification of stable maps) and enables us to prove the existence of rational smooth curves on homogeneous varities with fixed class in the Chow ring. This was not the case of Thomsen's and Kim and Pandharipande's proofs. We use a decomposition of G/P in orbits (called the P'-orbits, see definition) which are bigger than the Schubert cells. We then prove that these P'-orbits are "towers" of affine bundles (see definition) over "smaller" homogeneous varities. This description gives the results. Our decomposition in P'-orbits enables us to give a "better" desingularisation of Schubert varities than Demazure's one. | Courbes rationnelles sur les variétés homogènes et une
désingularisation plus fine des variétés de Schubert | 12,217 |
In their article [1], L. Gruson and M. Skiti have constructed a birationnal map from the variety $\I$ of mathematical instantons of degree 3 to the variety of nets of quadrics in $\pd$. They describe by this way two irreducible componants of the boundary of $\I$ associated to the divisor of nets which contain a two-plane degenerated quadric and the divisor of L\" uroth nets. In this article we describe an irreducible componante of the boundary of $\I$ as the exceptionnal divisor of the blowing-up of the closed set of nets of quadrics of rank 3. | Eclatement de réseaux de quadriques et bord des instantons de degré
3 | 12,218 |
We consider subtorus actions on complex toric varieties. A natural candidate for a categorical quotient of such an action is the so-called toric quotient, a universal object constructed in the toric category. We prove that if the toric quotient is weakly proper and if in addition the quotient variety is of expected dimension then the toric quotient is in fact a categorical quotient in the category of algebraic varieties. For example, weak properness always holds for the toric quotient of a subtorus action on a toric variety whose fan has a convex support. | Weakly Proper Toric Quotients | 12,219 |
An explicit upper bound for the Weil-Petersson volumes of the moduli spaces of punctured Riemann surfaces is obtained, using Penner's combinatorial integration scheme with embedded trivalent graphs. It is shown that for a fixed number of punctures n and for genus g going to infinity, the Weil-Petersson volume of M_{g,n} has an upper bound c^g g^{2g}. Here c is an independent of n constant, which is given explicitly. | Explicit upper bound for the Weil-Petersson volumes | 12,220 |
We prove that a smooth projective variety of dimension n is isomorphic to projective n-space iff the canonical class is -(n+1)-times an ample divisor. In characteristic zero this was proved by Kobayashi-Ochiai. We also extend the second adjunction theorem of Ionescu and Fujita to arbitrary characteristic. | Characterizations of ${\mathbb P}^n$ in arbitrary characteristic | 12,221 |
A fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones. Intersection homology, for example, directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety. Minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so. Here we use ideas from minimal model theory to define some characteristic numbers for singular varieties, generalizing the Chern numbers of a smooth variety. This was suggested by Goresky and MacPherson as a next natural problem after the definition of intersection homology. We find that only a subspace of the Chern numbers can be defined for singular varieties. A convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism, whereas a singular variety can at most have a fundamental class in a weaker homology theory, elliptic homology. We use this idea to give an algebro-geometric definition of elliptic homology: ``complex bordism modulo flops equals elliptic homology.'' | Chern numbers for singular varieties and elliptic homology | 12,222 |
The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call such a variety a Mori Dream Space. There turn out to be many examples, including quasi-smooth projective toric (or more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds. We characterize Mori dream spaces as GIT quotients of affine varieties by a torus in a manner generalizing Cox's construction of toric varieties as quotients of affine space. Via the quotient description, the chamber decomposition of the cone of divisors in Mori theory is naturally identified with the decomposition of the G-ample cone from geometric invariant theory. In particular every rational contraction of a Mori dream space comes from GIT, and all possible factorizations of a rational contraction can be read off from the chamber decomposition. | Mori Dream Spaces and GIT | 12,223 |
Let X be a projective variety which is covered by a family of rational curves of minimal degree. The classic bend-and-break argument of Mori asserts that if x and y are two general points, then there are at most finitely many curves in that family which contain both x and y. In this work we shed some light on the question as to whether two sufficiently general points actually define a unique curve. As an immediate corollary to the results of this paper, we give a characterization of projective spaces which improves on the known generalizations of Kobayashi-Ochiai's theorem. | Families of singular rational curves | 12,224 |
In this article we give a general approach to the following analogue of Shafarevich's conjecture for some polarized algebraic varieties; suppose that we fix a type of an algebraic variety and look at families of such type of varieties over a fixed Riemann surface with fixed points over which we have singular varieties, then one can ask if the set of such families, up to isomorphism, is finite. In this paper we give a general approach to such types of problems. The main observation is the following; suppose that the moduli space of a fixed type of algebraic polarized variety exists and suppose that in some projective smooth compactification of the coarse moduli the discriminant divisor supports an ample one, then it is not difficult to see that this fact implies the analogue of Shafarevich's conjecture. In this article we apply this method to certain polarized algebraic K3 surfaces and also to Enriques surfaces. | Ample Divisors, Automorphic Forms and Shafarevich's Conjecture | 12,225 |
In this paper we generalized the variational formulas for the determinants of the Laplacians on functions of CY metrics to forms of type (0,q) on CY manifolds. We also computed the Ray Singer Analytic torsion on CY manifolds we proved that it is bounded by a constant. In case of even dimensional CY manifolds the Ray Singer Analytic torsion is zero. The interesting case is the odd dimensional one. | Ray Singer Analytic Torsion of Calabi Yau manifolds I | 12,226 |
In this paper we construct the analogue of Dedekind eta function for odd dimensional CY manifolds. We use the theory of determinant line bundles. We constructed a canonical holomorphic section $\eta^{N}$ of some power of the determinant line bundle on the moduli space of odd dimensional CY manifolds. According to Viehweg the moduli space of moduli space of polarized odd dimensional CY manifolds $\mathcal{M}(M)$ is quasi projective. According to a Theorem due to Hironaka we can find a projective smooth variety $\bar {\mathcal{M}(M)}$ such that $\bar{\mathcal{M}(M)}\backslash$ $\mathcal{M}(M)=\mathcal{D}_{\infty}$ is a divisor of normal crossings. We also showed by using Mumford's theory of metrics with logarithmic growths that the determinant line bundle can be canonically prolonged to $\bar {\mathcal{M}(M)}$ $.$ We also showed that there exists section $\eta$ of some power of the determinant line bundle which vanishes on $\mathcal{D}_{\infty}$ and has a Quillen norm the Ray Singer Analytic Torsion$.$ This section is that analogue of the Dedekind eta Function $\eta.$ | Ray Singer Analytic Torsion of CY Manifolds II | 12,227 |
Let $(X,\Delta)$ be a 4-dimensional log variety which is proper over the field of complex numbers and with only divisorial log terminal singularities. The log canonical divisor $K_X+\Delta$ is semi-ample, if it is nef (numerically effective) and the Iitaka dimension $\kappa(X,K_X+\Delta)$ is strictly positive. For the proof, we use Fujino's abundance theorem for semi log canonical threefolds. | On Numerically Effective Log Canonical Divisors | 12,228 |
The goal of this note is to exhibit the integrability properties (in the sense of the Frobenius theorem) of holomorphic p-forms with values in certain line bundles with seminegative curvature on a compact Kaehler manifold. There are in fact very strong restrictions, both on the holomorphic form and on the curvature of the seminegative line bundle. As a consequence, these observations provide interesting information on the structure of projective manifolds which admit a contact structure: either they are Fano manifolds or, thanks to results of Kebekus-Peternell-Sommese-Wisniewski, they are biholomorphic to the projectivization of the cotangent bundle of another suitable projective manifold. | On the Frobenius integrability of certain holomorphic p-forms | 12,229 |
We compute the dimensions of spaces of sections of all powers of the Donaldson determinant bundle on the moduli space of rank 2 semi-stable sheaves on the projective plane, with zero first Chern class, and second Chern class equal to 3 or 4. We prove Le Potier's strange duality conjecture in some cases. | Resultats sur la dualite etrange sur le plan projectif | 12,230 |
Let X be a projective 3-fold with at most Q-factorial terminal singularities on which K_X is nef and big. Suppose the canonical index r(X)>1. For any positive integer m, it is interesting to consider the base point freeness and birationality of the divisor mK_X. For example, we know the following results: (1) the system |5rK_X| is base point free (Ein-Lazarsfeld-Lee); (2) |mK_X| gives a birational map for all m>4r+2 (M. Hanamura). This article aims to present a better result in direction (2). As far as our method can tell here, |mK_X| gives a birational map for all m>2r+5. (Q-divisor method + patient calculation) | Canonical stability in terms of singularity index for algebraic
threefolds | 12,231 |
We prove a few uniform versions of the Mordell-Lang Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points. The main focus is on function fields having high transcendence degree over the complex numbers. | Remarks about uniform boundedness of rational points over function
fields | 12,232 |
We show birationality of the morphism associated to line bundles $L$ of type $(1,...,1,2,...,2,4,...,4)$ on a generic $g-$dimensional abelian variety into its complete linear system such that $h^0(L)=2^g$. When $g=3$, we describe the image of the abelian threefold and from the geometry of the moduli space $SU_C(2)$ in the linear system $|2\theta_C|$, we obtain analogous results in $\p H^0(L)$. | Line bundles of type (1,,,1,2,,,2,4,,,4) on Abelian Varieties | 12,233 |
This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points. | Hurwitz numbers and intersections on moduli spaces of curves | 12,234 |
We give the multiplicity of terminal singularities on threefolds by simple calculation. Then we obtain the best inequalities for the multiplicity and the index. By using this, we can improve the boundedness number of terminal weak Q-Fano 3-folds in [KMMT, Theorem 1.2]. Furthermore, we can extended [K, Theorem 3.6] for Fujita freeness conditions to nonhypersurface terminal singularities. | On the multiplicity of terminal singularities on threefolds | 12,235 |
We describe the quantum cohomology rings of a class of toric varieties. The description includes, in addition to the (already known) ring presentations, the (new) analogues for toric varieties of the sorts of quantum Giambelli formulas which exist already for Grassmannian varieties, flag varieties, etc. | Gromov-Witten invariants of a class of toric varieties | 12,236 |
We show that polarisations of type (1,...,1,2g+2) on g-dimensional abelian varieties are $\it{never}$ very ample, if $g\geq 3$. This disproves a conjecture of Debarre, Hulek and Spandaw. We also give a criterion for non-embeddings of abelian varieties into 2g+1-dimensional linear systems. | Abelian Varieties into 2g+1-dim.Linear Systems | 12,237 |
For a linear system $|C|$ on a smooth projective surface $S$, whose general element is a smooth, irreducible curve, the Severi variety $V_{|C|, \delta}$ is the locally closed subscheme of $|C|$ which parametrizes irreducible curves with only $\delta$ nodes as singularities. In this paper we give numerical conditions on the class of divisors and upper-bounds on $\delta$ ensuring that the corresponding Severi variety is everywhere smooth of codimension $\delta$ in $|C|$ (regular, for short). In particular, we focus on surfaces of general type, since for such surfaces less is known than what is proven for other cases. Our result generalizes some results of Chiantini-Sernesi (1997) and of Greuel-Lossen-Shustin (1997 - in the case of nodes) as it is shown by some examples of Severi varieties on blown-up surfaces or surfaces in $\P^3$ which are elements of a component of the Noether-Lefschetz locus. We also consider examples of regular Severi varieties on surfaces in $\P^3$ of general type which contain a line. | Some results of regularity for Severi varieties of projective surfaces | 12,238 |
The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V.I.Arnold and V.A.Rokhlin. We make an attempt to systematize the principal results in the subject. After an exposition of general tools and results, special attention is paid to surfaces and curves on surfaces. | Topological properties of real algebraic varieties: du côté de chez
Rokhlin | 12,239 |
We prove that there exists a positive number $C_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over {\bf C}, the automorphism group $Aut(X)$ satisfies the inequality $\sharp{Aut}(X)\leq C_{n}\cdot\mu (X,K_{X})$, where $\mu (X,K_{X})$ is the volume of $X$ with respect to $K_{X}$. | Bound of automorphisms of projective varieties of general type | 12,240 |
We show that any birational map between projective hyperK\"ahler manifolds of dimension 4 is composed of a sequence of simple flops or elementary Mukai transformations under the assumption that each irreducible component of the indeterminacy of the birational map is normal. | HyperKähler Manifolds and Birational Transformations in dimension 4 | 12,241 |
Let $X\subset Y$ be smooth, projective manifolds. Assume that $X$ is the zero locus of a generic section of a direct sum $V+$ of positive line bundles on $\PP^n$. Furthermore assume that the normal bundle $N_{X/Y}$ is a direct sum $V-$ of negative line bundles. We show that a $V:=V+\oplus V-$-twisted Gromov-Witten theory of $\PP^n$ restricts to the Gromov-Witten theory of $X$ inherited form $Y$. The later one can be computed via a Mirror Theorem which we prove in this paper. | Mirror symmetry for concavex vector bundles on projective spaces | 12,242 |
The main result of this paper is that every (separated) toric variety which has a semigroup structure compatible with multiplication on the underlying torus is necessarily affine. In the course of proving this statement, we also give a proof of the fact that every separated toric variety may be constructed from a certain fan in a Euclidean space. To our best knowledge, this proof differs essentially from the ones which can be found in the literature. | On toric varieties and algebraic semigroups | 12,243 |
We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of ``affine lines'' of each degree in the complement of a smooth cubic in the projective plane. On the other hand, we consider coefficients of a certain expansion of a function obtained from the integrals of dxdy/xy over 2-chains whose boundaries lie on B_\phi where {B_\phi} is a family of smooth cubics. Then, for small degrees, they coincide. We discuss the relation between this phenomenon and local mirror symmetry for projective plane in a Calabi-Yau 3-fold by Chiang-Klemm-Yau-Zaslow. | Log mirror symmetry and local mirror symmetry | 12,244 |
We study the deformations of a holomorphic symplectic manifold $M$, not necessarily compact, over a formal ring. We show (under some additional, but mild, assumptions on $M$) that the coarse deformation space exists and is smooth, finite-dimensional and naturally embedded into $H^2(M)$. For a holomorphic symplectic manifold $M$ which satisfies $H^1(M,{\cal O}_M) = H^2(M,{\cal O}_M)=0$, the coarse moduli of formal deformations is isomorphic to $\Spec\C[[t_1, ..., t_n]]$, where $t_1$, ... $t_n$ are coordinates in $H^2(M)$. This revised version contains one minor improvement: exposition in Subsection 5.1 has been made more detailed and rigourous. | Period map for non-compact holomorphically symplectic manifolds | 12,245 |
Let M_g be the moduli space of smooth curves of genus g >= 3, and \bar{M}_g the Deligne-Mumford compactification in terms of stable curves. Let \bar{M}_g^{[1]} be an open set of \bar{M}_g consisting of stable curves of genus g with one node at most. In this paper, we determine the necessary and sufficient condition to guarantee that a Q-divisor D on \bar{M}_g is nef over \bar{M}_g^{[1]}, that is, (D . C) >= 0 for all irreducible curves C on \bar{M}_g with C \cap \bar{M}_g^{[1]} \not= \emptyset. | Nef divisors in codimension one on the moduli space of stable curves | 12,246 |
Let X be a smooth cubic threefold, M the moduli space of stable rank 2 vector bundles on X with trivial determinant and c_2=2 (the smallest value for which this space is non-empty). Recent results of Druel, Iliev, Markushevich and Tikhomirov lead to a precise description of M and its Maruyama compactification M': the latter is isomorphic to the intermediate Jacobian of X blown-up along the Fano surface. In this survey paper we explain these results and discuss some applications. | Vector bundles on the cubic threefold | 12,247 |
It was proved by Ginzburg and Mirkovic-Vilonen that the $G(O)$-equivariant perverse sheaves on the affine grassmannian of a connected reductive group $G$ form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group $G^\vee$. The proof use the Tannakian formalism. The purpose of this paper is to construct explicitely the action of $G^\vee$ on the global cohomology of a perverse sheaf. | On the action of the dual group on the cohomology of perverse sheaves on
the affine grassmannian | 12,248 |
We prove a conjecture of Frenkel, Gaitsgory, Kazhdan and Vilonen, related to Fourier coefficients of spherical perverse sheaves on the affine Grassmannian associated to a a split reductive group. Our proof is an extension of the proof given by the first author in the case of GL(n) (see math/9801109); it relies on the study of certain resolutions of Schubert varieties in the affine Grassmannian, built from the so-called minuscule or quasi-minuscule cases. | Résolutions de Demazure affines et formule de Casselman-Shalika
géométrique | 12,249 |
Let G be a semi-simple algebraic group over ${\mathbb C}$, B a Borel subgroup of G and T a maximal torus in B. A beautiful unpublished result of Dale Peterson says that if G is simply laced, then every rationally smooth point of a Schubert variety X in G/B is nonsingular in X. The purpose of this paper is to generalize this result to arbitrary T-stable subvarieties of G/B, the only restriction being that G contains no $G_2$ factors. In particular, we show that a Schubert variety X in such a G/B is nonsingular if and only if all the reduced tangent cones of X are linear. | On the Smooth Points of T-stable Varieties in G/B and the Peterson Map | 12,250 |
In this paper we prove that any smooth prime Fano threefold, different from the Mukai-Umemura threefold, contains a 1-dimensional family of intersecting lines. Combined with a result of the second author (see J. Algebr. Geom. 8:2 (1999), 221-244) this implies that any morphism from a smooth Fano threefold of index 2 to a smooth Fano threefold of index 1 must be constant, which gives an answer in dimension 3 to a question stated by Peternell. | Tangent scrolls in prime Fano threefolds | 12,251 |
Let $S_0$ be a smooth and compact real variety given by a reduced regular sequence of polynomials $f_1, ..., f_p$. This paper is devoted to the algorithmic problem of finding {\em efficiently} a representative point for each connected component of $S_0$ . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of $S_0$. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations $f_1, >..., f_p$ and in a suitably introduced, intrinsic geometric parameter, called the {\em degree} of the real interpretation of the given equation system $f_1, >..., f_p$. | Polar Varieties and Efficient Real Elimination | 12,252 |
Let $X$ be an ordinary smooth curve defined over an algebraically closed field of characteristic 2. The absolute Frobenius induces a rational map $F$ on the moduli space $M_X$ of rank 2 vector bundles with fixed trivial determinant. If the genus of $X$ is 2, the moduli space $M_X$ is isomorphic to projective space of dimension 3 (as over the complex numbers). In this case we explicitly give the equations of $F$, which enables us to determine, for example, its base locus (one point) and its image (different from $M_X$). | The action of the Frobenius map on rank 2 vector bundles in
characteristic 2 | 12,253 |
In this paper we address the following question arising from the work of P. Etingof, D. Kazhdan and A. Polishchuk (math.AG/0003009): given a homogeneous complex polynomial, when the rational map defined by its partials is of degree 1? We answer this question for reduced polynomials in three variables and also for reduced polynomials in four variables which factor into the product of linear polynomials. | Polar Cremona transformations | 12,254 |
On an affine variety $X$ defined by homogeneous polynomials, every line in the tangent cone of $X$ is a subvariety of $X$. However there are many other germs of analytic varieties which are not of cone type but contain ``lines'' passing through the origin. In this paper, we give a method to determine the existence and the ``number'' of such lines on non-degenerate surface singualrities. | Lines on Non-degenerate Surfaces | 12,255 |
Working in characteristic two, I classify nonsmooth Enriques surfaces with normal crossing singularities. Using Kato's theory of logarithmic structures, I show that such surfaces are smoothable and lift to characteristic zero, provided they are d-semistable. | Logarithmic deformations of normal crossing Enriques surfaces in
characteristic two | 12,256 |
Among solutions of n-Gelfand-Dikii's hierarchy there exists a remarkable solution W, which satisfies the string equation. We call it Witten's solution because according to the Witten conjecture the function F(x_1, x_2, x_3,...) = W(x_1,(x_2)/2, (x_3)/3,...) is the generating function for intersection nambers of Mumford-Morita-Muller cohomological classes of the moduli space of n-spin Riemann surfaces. This conjecture was proved by Kontsevich for n=2 and by Witten himself for surfaces of genus 0. In this paper we find recurrence relations between coefficients of Taylor series of W. This reduces the Witten's conjecture to conjecture that the Mamford-Morita-Muller numbers satisfy to the same relations. These relations give also an algorithm for calculation of $n$-spin Mamford-Morita-Muller numbers in assuming that the Witten conjecture is true. Moreover we prove that F(x_1,x_2,...,x_{n-1},0,0,...)= W(x_1, (x_2)/2,...,(x_{n-1})/(n-1),0,0,...). | Witten solution of the Gelfand-Dikii hierarchy | 12,257 |
Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas $\QQ$-Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric variety. | Homogeneous Coordinates and Quotient Presentations for Toric Varieties | 12,258 |
We prove the following version of Wlodarczyk's Embedding Theorem: Every normal complex algebraic ${\bf C}^*$-variety admits an equivariant closed embedding into a toric prevariety $X$ on which ${\bf C}^*$ acts as a one-parameter-subgroup of the big torus of $X$. | On Wlodarczyk's embedding theorem | 12,259 |
We characterize embeddability of algebraic varieties into smooth toric varieties and prevarieties. Our embedding results hold also in an equivariant context and thus generalize a well known embedding theorem of Sumihiro on quasiprojective G-varieties. The main idea is to reduce the embedding problem to the affine case. This is done by constructing equivariant affine conoids, a tool which extends the concept of an equivariant affine cone over a projective G-variety to a more general framework. | Equivariant embeddings into smooth toric varieties | 12,260 |
Let X be a smooth projective curve of genus g bigger then 2. For any vector bundle E on X let M_k(E) be the scheme of all rank k subbundles of E with maximal degree. For every integers r, k and x with 0<k<r, x positive and either x less then (k-1)(r-2k+1) (if 2k is less then r) or (r-k-1)(2k-r+1) (if 2k> r), we construct a rank r stable vector bundle E such that M_k(E) has an irreducible component of dimension x. Furthermore, if there exists a stable vector bundle F with small Lange's invariant s_k(F) and with M_k(F) `spread enough', then X i s a multiple covering of a curve of genus bigger then 2. | Families of maximal subbundles of stable vector bundles on curves | 12,261 |
We give a characterization of connected solvable groups in terms of the existence of representations with certain geometric properties. The existence of such representations for the group of upper triangular matrices played an important role in the proof of the authors' equivariant Riemann-Roch theorem. | Good representations and solvable groups | 12,262 |
In this paper we study the reduction of Galois covers of curves, from characteristic 0 to characteristic p. The starting point is a is a recent result of Raynaud which gives a criterion for good reduction for covers of the projective line branch at 3 points. Under some condition on the Galois group, we extend this criterion to the case of 4 branch points. Moreover, we describe the reduction of the Hurwitz space of such covers and compute the number of covers with good reduction. | Reduction of covers and Hurwitz spaces | 12,263 |
We give a complete description of the tautological subgroup of the fourth cohomology group of the moduli space of pointed stable curves, and prove that for g \geq 8 it coincides with the cohomology group itself. We further give a conjectural upper bound depending on the genus for the degree of new tautological relations. | The fourth tautological group of $\bar{\mathcal{M}_{g,n}}$ and relations
with the cohomology | 12,264 |
The purpose of this work is to describe the (category of) Higgs bundles on a complex scheme X having a given cameral cover X~. We show that this category is a T_{X~}-gerbe, where T_{X~} is a certain sheaf of abelian groups on X, and we describe the class of this gerbe precisely. In particular, it follows that the set of isomorphism classes of Higgs bundles with a fixed cameral cover X~ is a torsor over the group H^1(X, T_{X~}), which itself parametrizes T_{X~}-torsors on X. This underlying group can be described as a generalized Prym variety, whose connected component is either an abelian variety or a degeneration thereof. | The gerbe of Higgs bundles | 12,265 |
Homology with values in a connection with possibly irregular singular points on an algebraic curve is defined, generalizing homology with values in the underlying local system for a connection with regular singular points. Integration defines a perfect pairing between de Rham cohomology with values in the connection and homology with values in the dual connection. | Homology for irregular connections | 12,266 |
The paper consists of two sections. The first section provides a new definition of mirror symmetry of abelian varieties making sense also over $p$-adic fields. The second section introduces and studies quantized theta-functions with two-sided multipliers, which are functions on non-commutative tori. This is an extension of an earlier work by the author. In the Introduction and in the Appendix the constructions of this paper are put into a wider context. | Mirror symmetry and quantization of abelian varieties | 12,267 |
We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme X with given Hilbert polynomial h. This is a dg-manifold (smooth dg-scheme) RHilb_h(X) which carries a natural family of commutative (up to homotopy) dg-algebras, which over the usual Hilbert scheme is just given by truncations of the homogeneous coordinate rings of subschemes in X. In particular, RHilb_h(X) differs from RQuot_h(O_X), the derived Quot scheme constructed in our previous paper (math.AG/9905174) which carries only a family of A-infinity modules over the coordinate algebra of X. As an application, we construct the derived version of the moduli stack of stable maps of (variable) algebraic curves to a given projective variety Y, thus realizing the original suggestion of M. Kontsevich. | Derived Hilbert schemes | 12,268 |
On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line is a classical object and the curves are quantum. This generalizes to a new connection between algebraic geometry and the geometry of polyhedra, which is more straight-forward than the other known connections and gives a new insight into constructions used in the topology of real algebraic varieties. | Dequantization of real algebraic geometry on logarithmic paper | 12,269 |
We construct some natural cycles with trivial regulator in the higher Chow groups of Jacobians. For hyperelliptic curves we use a criterion due to J. Lewis to prove that the cycles we construct are indecomposable, and then use a specialization argument to prove indecomposability for more general curves. Please note that math.AG/9909062 has become an obsolete version of some of the results in the present paper. | Indecomposable Higher Chow cycles on Jacobians | 12,270 |
We study the degeneracy loci of general bundle morphisms from the direct sum of m copies of the structural sheaf on $P^n$ to $\Omega(2)$, also from the point of view of the classical geometrical interpretation of the sections of $\Omega(2)$ as linear line complexes. We consider in particular the case of $P^5$ with m=2, 3. For n=5 and m=3 we give an explicit description of the Hilbert scheme H of elliptic normal scrolls in $P^5$, by defining a natural rational map from the Grassmannian G(2,14) to H, which results to be dominant with general fibre of degree four. | On the construction of some Buchsbaum varieties and the Hilbert scheme
of elliptic scrolls in P^5 | 12,271 |
Let $X$ be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of $\kappa (X)$ in terms of the set $V^0(X,\omega_{X})$ $:=\{P\in {\text{\rm Pic}}^0(X)|h^0(X, \omega_X \otimes P) \ne 0\}$. An immediate consequence of this is that the Kodaira dimension $\kappa (X)$ is invariant under smooth deformations. We then study the pluricanonical maps $\phi_m:X -> \Bbb{P} (H^0(X,mK_X))$. We prove that if $X$ is of general type, $\phi_m$ is generically finite for $m\geq 5$ and birational for $m\geq 5 \text{\rm dim} (X) +1$. More generally, we show that for $m\geq 6$ the image of $\phi_m$ is of dimension equal to $\kappa (X)$ and for $m\geq 6\kappa (X)+2$, $\phi_m$ is the stable canonical map. | Pluricanonical maps of varieties of maximal Albanese dimension | 12,272 |
This note is a sequel to "Gerbes of chiral differential operators. II", math.AG/0003170. We study gerbes of chiral differential operators acting on the exterior algebra $\Lambda E$ of a vector bundle over a smooth algebraic variety $X$. When $E=\Omega^1_X$ this gerbe admits a canonical global section which coincides with the chiral de Rham complex of $X$. | Gerbes of chiral differential operators. III | 12,273 |
For an irreducible projective variety X, we study the family of h-planes contained in the secant variety Sec_k(X), for 0<h<k. These families have an expected dimension and we study varieties for which the expected dimension is not attained; for these varieties, making general consecutive projections to lower dimensional spaces, we do not get the expected singularities. In particular, we examine the family G of lines sitting in 3-secant planes to a surface S. We show that the actual dimension of G is equal to the expected dimension unless S is a cone or a rational normal scroll of degree 4 in P^5. | Grassmannians of secant varieties | 12,274 |
A scheme of computing $\chi(\mbar_{1,n}, L_1^{\otimes d_1}\otimes ... \otimes L_n^{\otimes d_n})$ is given. Here $\mbar_{1,n}$ is the moduli space of $n$-pointed stable curves of genus one and $L_i$ are the universal cotangent line bundles defined by $x_i^*(\omega_{C/M})$, where $C \to \mbar_{1,n}$ is the universal curve, $\omega_{C/M}$ the relative dualizing sheaf and $x_i$ the marked point. This work is a sequel to \cite{YL1}. | Orbifold Euler characteristics of universal cotangent line bundles on
$\bar{M}_{1,n}$ | 12,275 |
We introduce a category of extended complex manifolds, and prove that the functor describing deformations of a classical compact complex manifold $M$ within this category is versally representable by (an analytic subspace in) $H^*(M,T_M)$. By restricting the associated versal family of extended complex manifolds over $H^*(M,T_M)$ to the subspace $H^1(M,T_M)$ one gets a correct limit to the classical picture. | A note on extended complex manifolds | 12,276 |
Let HH_{ab}(H) be the equivariant Hilbert scheme parametrizing the zero dimensional subschemes of the affine plane k^2, fixed under the one dimensional torus T_{ab}={(t^{-b},t^a), t\in k^*} and whose Hilbert function is H. This Hilbert scheme admits a natural stratification in Schubert cells which extends the notion of Schubert cells on Grassmannians. However, the incidence relations between the cells become more complicated than in the case of Grassmannians. In this paper, we give a necessary condition for the closure of a cell to meet another cell. In the particular case of Grassmannians, it coincides with the well known necessary and sufficient incidence condition. There is no known example showing that the condition wouldn't be sufficient. | Incidence relations among the Schubert cells of equivariant Hilbert
Schemes | 12,277 |
The Witt group of a smooth curve over a real closed field is explicitely calculated. The method uses a comparison theorem between the graded Witt group and the etale cohomology groups. In the second part of the paper, the torsion Picard group of a smooth real curve is also computed. The calculation depends on a new invariant introduced here. We prove that several problems of real algebraic geometry are related to this invariant. | Witt groups and torsion Picard groups of smooth real curves | 12,278 |
Let X be a smooth complex surface of general type such that the image of the canonical map $\phi$ of X is a surface $\Sigma$ and that $\phi$ has degree $\delta\geq 2$. Let $\epsilon\colon S\to \Sigma$ be a desingularization of $\Sigma$ and assume that the geometric genus of S is not zero. Beauville has proved that in this case S is of general type and $\epsilon$ is the canonical map of S. Beauville has also constructed the only infinite series of examples $\phi:X\to \Sigma$ with the above properties that was known up to now. Starting from his construction, we define a {\em good generating pair}, namely a pair $(h:V\to W, L)$ where h is a finite morphism of surfaces and L is a nef and big line bundle of W satisfying certain assumptions. We show that by applying a construction analogous to Beauville's to a good generating pair one obtains an infinite series of surfaces of general type whose canonical map is 2-to-1 onto a canonically embedded surface. In this way we are able to construct more infinite series of such surfaces. In addition, we show that good generating pairs have bounded invariants and that there exist essentially only 2 examples with $\dim |L|>1$. The key fact that we exploit for obtaining these results is that the Albanese variety P of V is a Prym variety and that the fibre of the Prym map over P has positive dimension. | Prym varieties and the canonical map of surfaces of general type | 12,279 |
The aim of this paper is to obtain a classification of scrolls of genus 0 and 1, which are defined by a one-dimensional family of lines meeting a certain set of linear spaces in ${\bf P}^n$. These ruled surfaces will be called incidence scrolls and such a set will be the base of the incidence scroll. Unless otherwise stated, we assume that the base spaces are in general position. | Classification of Incidence Scrolls(I) | 12,280 |
Let $X$ be a compact K\"{a}hler manifold, and let $L$ be a line bundle on $X.$ Define $I_k(L)$ to be the kernel of the multiplication map $ Sym^k H^0 (L) \to H^0 (L^k).$ For all $h \leq k,$ we define a map $\rho : I_k(L) \to Hom (H^{p,q} (L^{-h}), H^{p+1,q-1} (L^{k-h})).$ When $L = K_X$ is the canonical bundle, the map $\rho$ computes a second fundamental form associated to the deformations of $X.$ If $X=C$ is a curve, then $\rho$ is a lifting of the Wahl map $I_2(L) \to H^0 (L^2 \otimes K_C^2).$ We also show how to generalize the construction of $\rho$ to the cases of harmonic bundles and of couples of vector bundles. | Hodge-Gaussian maps | 12,281 |
In this paper, we introduce the notion of generalized rational Okamoto-Painlev\'e pair (S, Y) by generalizing the notion of the spaces of initial conditions of Painlev\'e equations. After classifying those pairs, we will establish an algebro-geometric approach to derive the Painlev\'e differential equations from the deformation of Okamoto-Painlev\'e pairs by using the local cohomology groups. Moreover the reason why the Painlev\'e equations can be written in Hamiltonian systems is clarified by means of the holomorphic symplectic structure on S - Y. Hamiltonian structures for Okamoto-Painlev\'e pairs of type $\tilde{E}_7 (= P_{II})$ and $\tilde{D}_8 (= P_{III}^{\tilde{D}_8})$ are calculated explicitly as examples of our theory. | Deformation of Okamoto-Painlevé Pairs and Painlevé equations | 12,282 |
In the theory of deformation of Okamoto-Painlev\'e pair (S,Y), a local cohomology group $H^1_D(\Theta_S(-\log D))$ plays an important role. In this paper, we estimate the local cohomology group of pair (S,Y) for several types, and obtain the following results. For a pair (S,Y) corresponding to the space of initial conditions of the Painlev\'e equations, we show that the local cohomology group $H^1_D(\Theta_S(-\log D))$ is at least 1 dimensional. This fact is the key to understand Painlev\'e equation related to (S,Y). Moreover we show that, for the pairs (S,Y) of type $\tilde{A_8}$, the local cohomology group $H^1_D(\Theta_S(-\log D))$ vanish. Therefore in this case, there is no differential equation on S-Y in the sense of the theory. | Local cohomology of generalized Okamoto-Painlevé pairs and Painlevé
equations | 12,283 |
In this paper, we introduce the notion of an Okamoto-Painlev\'e pair (S, Y) which consists of a compact smooth complex surface S and an effective divisor Y on S satisfying certain conditions. Though spaces of initial values of Painlev\'e equations introduced by K. Okamoto give examples of Okamoto-Painleve pairs, we find a new example of Okamoto-Painlev\'e pairs not listed in \cite{Oka}. We will give the complete classification of Okamoto-Painlev\'e pairs. | Classification of Okamoto-Painlevé Pairs | 12,284 |
Let $X$ be a smooth projective curve over the complex numbers. To every representation $\rho\colon \GL(r)\lra \GL(V)$ of the complex general linear group on the finite dimensional complex vector space $V$ which satisfies the assumption that there be an integer $\alpha$ with $\rho(z \id_{\C^r})=z^\alpha \id_V$ for all $z\in\C^*$ we associate the problem of classifying triples $(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ on $X$, $L$ is a line bundle on $X$, and $\phi\colon E_\rho\lra L$ is a non trivial homomorphism. Here, $E_\rho$ is the vector bundle of rank $\dim V$ associated to $E$ via $\rho$. If we take, for example, the standard representation of $\GL(r)$ on $\C^r$ we have to classify triples $(E,L,\phi)$ consisting of $E$ as before and a non-zero homomorphism $\phi\colon E\lra L$ which includes the so-called Bradlow pairs. For the representation of $\GL(r)$ on $S^2\C^3$ we find the conic bundles of Gomez and Sols. In the present paper, we will formulate a general semistability concept for the above triples which depends on a rational parameter $\delta$ and establish the existence of moduli spaces of $\delta$-(semi)stable triples of fixed topological type. The notion of semistability mimics the Hilbert-Mumford criterion for $SL(r)$ which is the main reason that such a general approach becomes feasible. In the known examples (the above, Higgs bundles, extension pairs, oriented framed bundles) we show how to recover the "usual" semistability concept. This process of simplification can also be formalized. Altogether, our results provide a unifying construction for the moduli spaces of most decorated vector bundle problems together with an automatism for finding the right notion of semistability and should therefore be of some interest. | A universal construction for moduli spaces of decorated vector bundles
over curves | 12,285 |
To every oriented tree, we associate a moduli problem for sheaves over a projective manifold $X$. We define the corresponding notion of semistability and establish the existence of moduli spaces. Applying the results to the tree *->*, we obtain a GIT construction for the moduli space of holomorphic triples of Bradlow and Garcia-Prada. | Moduli problems of sheaves associated with oriented trees | 12,286 |
This is an erratum to math.AG/9803126, Tohoku 51 (1999) 489-537. This erratum describes: 1. the failure of the algorithm in [AMR] and [Morelli1] for the strong factorization pointed out by Kalle Karu, 2. the statement of a refined weak factorization theorem for toroidal birational morphisms in [AMR], in the form utilized in [AKMR] for the proof of the weak factorization theorem for general birationla maps, avoiding the use of the above mentioned algorithm for the strong factorization, and 3. a list of corrections for a few other mistakes in [AMR], mostly pointed out by Laurent Bonavero. | Erratum to the paper "A note on the factorization theorem of toric
birational maps after Morelli and its toroidal extension" | 12,287 |
This paper is originally designed as a part of revision of the author's preprint math.AG/9908174 "P-adic Schwarzian triangle groups of Mumford type". Recently, Yves Andr'e pointed out a flaw in that preprint; more precisely, Proposition II there was not correct, and consequently, as he pointed out, some triangle groups are missing (but only in p=2,3,5). I apologize to the readers, and will release the other part soon by the original title. In this paper, we concentrate on the constructive part, but more systematically and in more general situation. More precisely, this paper presents a complete criterion for an abstract tree of groups to be realized in the context of p-adic uniformization. This criterion provides a practical way of constructing finitely generated discrete subgroups in PGL(2,K) over a p-adic field K. In the end of this paper, we will exhibit concrete examples of such construction, which in particular shows that there exist infinitely many triangle groups. | Graph Theoretic Construction of Discrete Groups over p-adic Fields | 12,288 |
We describe the moduli spaces of morphisms between polarized complex abelian varieties. The discrete invariants, derived from a Poincare' decomposition of morphisms, are the types of polarizations and of lattice homomorphisms occurring in the decomposition. For a given type of morphisms the moduli variety is irreducible, and is obtained from a product of Siegel spaces modulo the action of a discrete group. | Siegel coordinates and moduli spaces for morphisms of abelian varieties | 12,289 |
Let G be a complex reductive group and let C be a smooth curve of genus at least one. We prove a converse to a theorem of Atiyah-Bott concerning the stratification of the space of holomorphic G-bundles on C. In case the genus of C is one, we establish that one has a stratification in the strong sense. The paper concludes with a characterization of the minimally unstable strata in case G is simple. | On the converse to a theorem of Atiyah and Bott | 12,290 |
In this short note, we determine the Kodaira dimension and some of the plurigenera of (a desingularization of) a symmetric power of a smooth projective variety. We use it to obtain bounds on the genus of curve passing through a fixed number of general points. | Kodaira dimension of symmetric powers | 12,291 |
Let V be a convex vector bundle over a smooth projective manifold X, and let Y be the subset of X which is the zero locus of a regular section of V. This mostly expository paper discusses a conjecture which relates the virtual fundamental classes of X and Y. Using an argument due to Gathmann, we prove a special case of the conjecture. The paper concludes with a discussion of how our conjecture relates to the mirror theorems in the literature. | Virtual Fundamental Classes of Zero Loci | 12,292 |
Simplicial complexes X provide commutative rings A(X) via the Stanley-Reisner construction. We calculated the cotangent cohomology, i.e., T1 and T2 of A(X) in terms of X. These modules provide information about the deformation theory of the algebro geometric objects assigned to X. | Cotangent cohomology of Stanley-Reisner rings | 12,293 |
We study Deligne's conjecture on the monodromy weight filtration on the nearby cycles in the mixed characteristic case, and reduce it to the nondegeneracy of certain pairings in the semistable case. We also prove a related conjecture of Rapoport and Zink which uses only the image of the Cech restriction morphism, if Deligne's conjecture holds for a general hyperplane section. In general we show that Deligne's conjecture is true if the standard conjectures hold. | Monodromy Filtration and Positivity | 12,294 |
This is a sequal paper to math.AG/9909021. By using the theory of AZD originated by the author, I prove that for every smooth projective $n$-fold $X$ of general type and every \[ m\geq \lceil\sum_{\ell =1}^{n}\sqrt[\ell]{2} \ell\rceil +1, \] $\mid mK_{X}\mid$ gives a birational rational map from $X$ into a projective space, unless it has a nontrivial (relative dimension is positive) rational fiber space structure whose general fiber is birational to a variety of relatively low degree in a projective space. | Effective birationality of pluricanonical systems | 12,295 |
If a complex analytic function, $f$, has a stratified isolated critical point, then it is known that the cohomology of the Milnor fibre of $f$ has a direct sum decomposition in terms of the normal Morse data to the strata. We use microlocal Morse theory to obtain the same result under the weakened hypothesis that the vanishing cycles along $f$ have isolated support. We also investigate an index-theoretic proof of this fact. | A Little Microlocal Morse Theory | 12,296 |
We introduce the notion of lef line bundles on a complex projective manifold. We prove that lef line bundles satisfy the Hard Lefschetz Theorem, the Lefschetz Decomposition and the Hodge-Riemann Bilinear Relations. We study proper holomorphic semismall maps from complex manifolds and prove that, for constant coefficients, the Decomposition Theorem is equivalent to the non-degeneracy of certain intersection forms. We give a proof of the Decomposition Theorem for the complex direct image of the constant sheaf when the domain and the target are projective by proving that the forms in question are non-degenerate. A new feature uncovered by our proof is that the forms are definite. | The Hard Lefschetz Theorem and the topology of semismall maps | 12,297 |
In this note, we study monodromies of algebraic connections on the trivial vector bundle. We prove that on a smooth complex affine curve, any monodromy arises as the underlying local system of an algebraic connection on the trivial bundle. We give a generalization of this for rank one monodromies in higher dimension. | Monodromies of Algebraic Connections on the Trivial Bundle | 12,298 |
A tetragonal canonical curve is the complete intersection of two divisors on a scroll. The equations can be written in `rolling factors' format. For such homogeneous ideals we give methods to compute infinitesimal deformations. Deformations can be obstructed. For the case of quadratic equations on the scroll we derive explicit base equations. They are used to study extensions of tetragonal curves. | Rolling Factors Deformations and Extensions of Canonical Curves | 12,299 |
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