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At least 163 records · Page 9Linked to original sources
arXiv2001

Hyperelliptic jacobians and $\U_3(2^m)$

In his previous paper (Math. Res. Letters 7(2000), 123--132) the author proved that in characteristic zero the jacobian $J(C)$ of a hyperelliptic curve $C: y^2=f(x)$ has only trivial endomorphisms over an algebraic closure $K_a$ of the ground field $K$ if the Galois group $Gal(f)$ of the irreducible polynomial $f(x) \in K[x]$ is either the symmetric group $S_n$ or the alternating group $A_n$. Here $n>4$ is the degree of $f$. In math.AG/0003002 we extended this result to the case of certain ``smaller'' Galois groups. In particular, we treated the infinite series $n=2^r+1, Gal(f)=L_2(2^r)$ and $n=2^{4r+2}+1, Gal(f)=Sz(2^{2r+1})$. In this paper we do the case of $Gal(f)=\U_3(2^m)$ and $n=2^{3m}+1$.

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arXiv2001

Using symmetry to count rational curves

An analogy is drawn between recent work with Kley (math.AG/0007082) and the WDVV equations. That is, both are regarded as symmetries of generating functions with coefficients that "count" rational curves on a complex projective manifold. It is shown how to obtain the string, dilaton and divisor equations from the new symmetries, as well as an analogue of the Kontsevich-Manin reconstruction theorem, expressing arbitrary genus zero Gromov-Witten invariants in terms of "mirror data". As was also pointed out in the work with Kley, this allows one, for the first time, to compute quantum cohomology from mirror data.

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arXiv2001

Curves of maximal genus in P^5

Let C be a reduced, irreducible, not degenerate curve, not contained on surfaces of degree 3, the bound is stated and proved by Chiantini, Ciliberto and Di Gennaro in 1993. The bound is sharp, at least for d sufficiently large (Chiantini-Ciliberto- Di Gennaro give examples of extremal curves for d large, that are in general singular). The classification and the existence of curves of genus G(d, r, s) is known for r=3 and d>s^2-s (Gruson and Peskine), and for r=4 and d> 12s^2 (Chiantini and Ciliberto in 1994). In this paper the author gives the classification for the curves of maximal genus for r=5, d as in the paper of Chiantini-Ciliberto-Di Gennaro, and s>8, and proves the existence of {\it smooth} curves of maximal genus G(d, 5, s) for every d and s. With the same techniques used for the classification in P^5 it is possible to classify curves in P^r of maximal genus G(d, r, s) for every r and s>2r-2. In math.AG/0105094 the author has given an example of the classification procedure and of the costruction of smooth extremal curves in P^r.

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arXiv2001

From coherent sheaves to curves in {\bf P}^3

This paper is the text of a lecture given at the Catania conference on Commutative Algebra and Algebraic Geometry, in honor of the 60th birthday of Silvio Greco. It analyses the correspondences between equivalence classes of objects related to the projective 3-space ${\bf P}^3_k$ over a field $k$ : - vector bundles ${\cal F}$ such that $H_*^2({\cal F})=0$ up to stable equivalence - coherent sheaves of projective dimension $\leq 1$ up to pseudo-equivalence - finite length graded $k[X,Y,Z,T]$-modules, modulo isomorphism - biliaison classes of locally Cohen-Macaulay curves. In particular, it leads to the natural question : ``If a biliaison class contains subcanonical curves, is a minimal curve subcanonical ?'' for which a positive answer was given in math.AG/0102222.

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arXiv2001

Twisted bundles and admissible covers

The cherry on top of this stacky paper is the following: for any g>1 we give a finite group G such that the moduli space of connected admissible G-covers of genus g is a smooth, fine moduli space, which is a Galois cover of the moduli space of stable curves. The proof relies on methods introduced by Looijenga and Pikaart-De Jong, and on the theory of twisted G-covers, a theory announced without proofs in math.AG/9811059, section 3, and developed in the first few sections of this paper. This includes a moduli description of the normalization of the Harris-Mumford space of admissible covers, and a study of moduli of stable curves with abelian and non-abelian level structure.

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arXiv2001

Multiple quantum products in toric varieties

We generalize the author's formula for Gromov-Witten invariants of symplectic toric manifolds (see math.AG/0006156) to those needed to compute the quantum product of more than two classes directly, i.e. involving the pull-back of the Poincaré dual of the point class in the Deligne-Mumford spaces $ \mathcal{M}_{0,m} $.

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arXiv2001

$π_1$-classification of real arrangements with up to eight lines

One of the open questions in the geometry of line arrangements is to what extent does the incidence lattice of an arrangement determine its fundamental group. Line arrangements of up to 6 lines were recently classified by K.M. Fan, and it turns out that the incidence lattice of such arrangements determines the projective fundamental group. We use actions on the set of wiring diagrams, introduced in [GTV] (math.AG/0107178), to classify real arrangements of up to 8 lines. In particular, we show that the incidence lattice of such arrangements determines both the affine and the projective fundamental groups.

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arXiv2001

The Toda conjecture

We study the Toda conjecture of Eguchi and Yang for the Gromov-Witten invariants of CP^1,using the bihamiltonian method of the formal calculus of variations. We also study its relationship to the Virasoro conjecture for CP^1, recently proved by Givental (math.AG/0108100).

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arXiv2003

Regularity on abelian varieties II: basic results on linear series and defining equations

This paper is mainly concerned with applying the theory of M-regularity developed in the previous math.AG/0110003 to the study of linear series given by multiples of ample line bundles on abelian varieties. We define a new invariant of a line bundle, called M-regularity index, which is seen to govern the higher order properties and (partly conjecturally) the defining equations of such embeddings. We prove a general result on the behavior of higher syzygies in embeddings given by multiples of ample bundles whose base locus has no fixed components, extending a conjecture of Lazarsfeld known to be true by work of the first author. This approach also unifies essentially all the previously known results in this direction and is mainly based on vanishing theorems rather than representations of theta groups.

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arXiv2001

A Degeneration formula of GW-invariants

This is the second part of the paper "A degeneration of stable morphisms and relative stable morphisms", (math.AG/0009097). In this paper, we constructed the relative Gromov-Witten invariants of a pair of a smooth variety and a smooth divisor. We then proved a degeneration formula of Gromov-Witten invariants, in cycle form.

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arXiv2003

Full faithfulness for overconvergent F-isocrystals

Let X be a smooth variety over a field of characteristic p>0. We prove that the forgetful functor from the category of overconvergent F-isocrystals on X to the category of convergent F-isocrystals is fully faithful. The argument uses the quasi-unipotence theorem for overconvergent F-isocrystals (recently proved independently by Andre, Mebkhout, and the author; see math.AG/0110124), plus arguments of de Jong. In the process, we establish a theorem of Quillen-Suslin type (i.e., every finite projective module is free) over rings of overconvergent power series.

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arXiv2005

Rank 4 vector bundles on the quintic threefold

By the results of the author and Chiantini in Math.AG/0110102, on a general quintic threefold $X \subset {\mathbf P}^4$ the minimum integer $p$ for which there exists a positive dimensional family of irreducible rank $p$ vector bundles on $X$ without intermediate cohomology is at least three. In this paper we show that $p \leq 4$, by constructing series of positive dimensional families of rank 4 vector bundles on $X$ without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class $Ext^1(E,F)$, for a suitable choice of the rank 2 ACM bundles $E$ and $F$ on $X$. The existence of such bundles of rank $p = 3$ remains under question.

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arXiv2001

The Chow ring of a K3 surface

Let X be a K3 surface. We show that the Chow group CH_0(X) of 0-cycles contains a "fundamental class" c_X of degree 1 with remarkable properties: any product of divisors is proportional to this class, and so is the second Chern class c_2(X). This preprint supersedes the preprint math.AG/0109063 by the first author, where the latter property was conjectured.

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arXiv2001

Kustin-Miller unprojection with complexes

A main ingredient for Kustin-Miller unprojection, as developed in (S. Papadakis and M. Reid, Kustin-Miller unprojection without complexes, math.AG/0011094), is the module Hom_R(I, \om_R), where R is a local Gorenstein ring and I a codimension one ideal with R/I Gorenstein. We prove a method of calculating it in a relative setting using resolutions. We give three applications. In the first we generalise a result of (F. Catanese et al., Embeddings of curves and surfaces, Nagoya Math. J. 154 (1999), 185-220). The second and the third are about Tom and Jerry, two families of Gorenstein codimension four rings with 9x16 resolutions.

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arXiv2001

Puzzles and (equivariant) cohomology of Grassmannians

We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in {y_{i+1} - y_i}; they were shown (abstractly) to have positive coefficients in [Graham] math.AG/9908172. Our formula is the first to be manifestly positive in this sense. In particular this gives a new and self-contained proof of the ordinary puzzle formula, by an induction backwards from the "most equivariant" case. The proof of the formula is mostly combinatorial, but requires no prior combinatorics, and only a modicum of equivariant cohomology (which we include). This formula is closely related to the one in [Molev-Sagan] q-alg/9707028 for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [Graham]. We include a cohomological interpretation of this problem, and a puzzle formulation for it.

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arXiv2001

Limites de fibres vectoriels dans ${\bf M}_{\mathbb{Q}_3}(0,2,0)$

Let $\mathbb{Q_3}$ the smooth quadric in P^4 (the 4-dimensional projective space). In this note we completely describe the closure of the open set of vector bundles in ${\bf M}_{\mathbb{Q}_3}(0,2,0)$. This description gives another example where the conditions given in math.AG/0112199 for a sheaf to be limit of vector bundles are necessary and sufficient.

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