arXiv subjects

Eduardo Esteves

Publications and source records attributed to Eduardo Esteves.

41 records · Page 3Linked to original sources

arXiv1999

Autoduality of the compactified Jacobian

We prove the following autoduality theorem for an integral projective curve C in any characteristic. Given an invertible sheaf L of degree 1, form the corresponding Abel map A_L: C->J, which maps C into its compactified Jacobian, and form its pullback map A_L^*: Pic^0_J to J, which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, points of multiplicity 2, then A_L^* is an isomorphism, and forming it commutes with specializing C. Much of our work is valid, more generally, for a family of curves with, at worst, points of embedding dimension 2. In this case, we use the determinant of cohomology to construct a right inverse to A_L^*. Then we prove a scheme-theoretic version of the theorem of the cube, generalizing Mumford's, and use it to prove that A_L^* is independent of the choice of L. Finally, we prove our autoduality theorem: we use the presentation scheme to achieve an induction on the difference between the arithmetic and geometric genera; here, we use a few special properties of points of multiplicity 2.

math.AG↗

arXiv1998

Linear systems and ramification points on reducible nodal curves

In the 80's D. Eisenbud and J. Harris developed the general theory of limit linear series, Invent. math. 85 (1986), in order to understand what happens to linear systems and their ramification points on families of non-singular curves degenerating to curves of compact type. They applied their theory to the study of limits of Weierstrass points, among other endeavours. In one of their articles, Invent. math. 87 (1987), they asked: "What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type?" In this eprint we address this question within a more general framework. More precisely, given a family of linear systems on a family of non-singular curves degenerating to a nodal curve we give a formula for the limit of the associated ramification divisors in terms of certain limits of the family of linear systems. In contrast with the theory of limit linear series of Eisenbud's and Harris', we do not need to blow up the family to swerve the degenerating ramification points away from the nodes of the limit curve. Indeed, we can assign the adequate weight to the limit ramification divisor at any point of the limit curve. In a forthcoming submission we shall deal with the specific question of limits of Weierstrass points, assuming certain generic conditions.

math.AG↗

arXiv1997

Separation properties of theta functions

In a 1993 article, G. Faltings gave a new construction of the moduli space $U$ of semistable vector bundles on a smooth curve $X$, avoiding geometric invariant theory. Roughly speaking, Faltings showed that the normalisation $B$ of the ring $A$ of theta functions (associated with vector bundles on $X$) suffices to realize $U$ as a projective variety. Describing Faltings' work, C.S. Seshadri asked how close $A$ is to $B$. In this article, we address this question from a geometric point of view. We consider the rational map, $π: U @>>> Proj(A)$, and show that, not only is $π$ defined everywhere, but also $π$ is bijective, and is an isomorphism over the stable locus of $U$, if the characteristic of the ground field is 0. Moreover, we give a direct local construction of $U$ as a fine moduli space, when the rank and degree are coprime, in any characteristic. The methods in the article apply to singular curves as well.

alg-geom↗

arXiv1997

Compactifying the relative Jacobian over families of reduced curves

We construct natural relative compactifications for the relative Jacobian over a family $X/S$ of reduced curves. In contrast with all the available compactifications so far, ours admit a universal sheaf, after an etale base change. Our method consists of considering the functor $F$ of relatively simple, torsion-free, rank 1 sheaves on $X/S$, and showing that certain open subsheaves of $F$ have good properties. Strictly speaking, the functor $F$ is only representable by an algebraic space, but we show that $F$ is representable by a scheme after an etale base change. Finally, we use theta functions originating from vector bundles to compare our new compactifications with the available ones.

alg-geom↗

arXiv1997

Very ampleness for Theta on the compactified Jacobian

The Jacobian $J$ of a complete, smooth, connected curve $X$ admits a canonical divisor $Θ$, called the Theta divisor. It is well-known that $Θ$ is ample and, in fact, $3Θ$ is very ample. For a general complete, integral curve $X$, D'Souza constructed a compactification $\bar J$ of the Jacobian $J$ by considering torsion-free, rank 1 sheaves on $X$. Soucaris and the author considered independently the analogous Theta divisor $Θ$ on $\bar J$, and showed that $Θ$ is ample. In this article, we show that $nΘ$ is very ample for $n$ greater or equal to a specified lower bound. If $X$ has at most ordinary nodes or cusps as singularities, then our lower bound is 3. Our main tool is to use theta sections associated to vector bundles on $X$ to embed $\bar J$ into a projective space.

alg-geom↗