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Lucia Caporaso

Publications and source records attributed to Lucia Caporaso.

At least 37 records · Page 2Linked to original sources
arXiv2010

Linear series on semistable curves

The dimension of spaces of global sections for line bundles on semistable curves parametrized by the compactified Picard scheme is studied. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: the curve has two components; the curve is any semistable curve and the degree is either 0 or 2g-2; the curve is stable, free from separating nodes, and the degree is at most 4. These results are all shown to be sharp. Applications to the Clifford index, to the combinatorial description of hyperelliptic curves, and to plane quintics are given.

math.AG↗
arXiv2010

Torelli theorem for stable curves

We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs. We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper bound on the cardinality of the finite fibers. We also bound the dimension of the infinite fibers. In particular, we obtain that the compactified Torelli map is injective away from curve with separating nodes if and only if the genus is at most 4.

math.AG↗
arXiv2009

Torelli theorem for graphs and tropical curves

Algebraic curves have a discrete analogue in finite graphs. Pursuing this analogy we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for 3-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. Finally we describe some natural posets associated to a graph and prove that they characterize its Delaunay decomposition.

math.CO↗
arXiv2008

Geometry of the theta divisor of a compactified jacobian

We study the theta divisor of the compactified jacobian of a nodal, possibly reducible, curve. We compute its irreducible components and give it a geometric interpretation consistent with the classical Brill-Noether theory of smooth curves. Some applications on hyperelliptic stable curves are appended.

math.AG↗
arXiv2007

Abel maps of Gorenstein curves

For a Gorenstein curve X and a nonsingular point P of X, we construct Abel maps A from X to J_X^1 and A_P from X to J_X^0, where J_X^i is the moduli scheme for simple, torsion-free, rank-1 sheaves on X of degree i. The image curves of A and A_P are shown to have the same arithmetic genus of X. Also, A and A_P are shown to be embeddings away from rational subcurves L of X meeting the closure of X-L in separating nodes. Finally, we establish a connection with Seshadri's moduli scheme U_X(1) for semistable, torsion-free, rank-1 sheaves on X, obtaining an embedding of A(X) into U_X(1).

math.AG↗
arXiv2007

Moduli of roots of line bundles on curves

We treat the problem of completing the moduli space for roots of line bundles on curves. Special attention is devoted to higher spin curves within the universal Picard scheme. Two new different constructions, both using line bundles on nodal curves as boundary points, are carried out and compared with pre-existing ones.

math.AG↗
arXiv2006

Naturality of Abel maps

We give a combinatorial characterization of nodal curves admitting a natural d-th Abel map to their Picard scheme, for any positive integer d. "Natural" here means compatible with and independent of specialization.

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arXiv2006

Neron models and Compactified Picard schemes over the moduli stack of stable curves

We prove that there exist some stacks, representable over the stack of stable curves, having the following universal property with respect to Néron models of Jacobians. For every one-parameter family of stable curves, with regular total space, the Néron model of the Jacobian of its generic fibre is isomorphic to the base change of the above stacks via the moduli map of the given family. We also obtain a stack compactification of the universal Picard scheme and hence a geometrically meaningful completion of the Néron model of the Jacobian for every family as above.

math.AG↗
arXiv2006

On Abel maps of stable curves

We construct Abel maps for a stable curve $X$. Namely, for each one-parameter deformation of $X$ with regular total space, and every integer $d>0$, we construct by specialization a map $α^d_X$ from the smooth locus of $X^d$ to the moduli scheme of balanced line bundles on semistable curves over $X$. For $d=1$, we show that $α^1_X$ naturally extends over $X$, and does not depend on the choice of the deformation; we give a precise description of when it is injective.

math.AG↗
arXiv2002

Combinatorial properties of stable spin curves

The geometry of the moduli space of stable spin curves is studied, with emphasis on its combinatorial properties. In this context, the standard graph theoretic framework is not just a book-keeping device: some purely combinatorial results are proved, having moduli theoretic applications. In particular, certain strata of the moduli space of stable curves are characterized by a (finite) set of integers, measuring the non-reducedness of the scheme of spin curves, and definable in purely graph-theoretical terms.

math.AG↗
arXiv2002

Characterizing Curves by their odd Theta-characteristics

We show that a general canonical curve is uniquely determined by the finite set of hyperplanes cutting theta-characteristics on it. Geometrical and combinatorial properties of the moduli space of stable spin curves are proved, which play an essential role.

math.AG↗
arXiv2001

Recovering plane curves from their bitangents

We show that a general plane curve of degree at least 4 is uniquely determined by the full set of its bitangent lines. This problem has an elementary solution for degree at least 5, and the paper is almost entirely devoted to curves of degree 4, where we generalize the result to nodal quartics. In other words, we show that a general curve of genus 3 can be recovered from its 28 odd theta-characteristics.

math.AG↗
arXiv2000

On modular properties of odd theta-characteristics

A general canonical curve X determines a finite set T(X) of hyperplanes, which is in bijective correspondence with the set of odd theta-characteristics of X. The definition of T(X) can be extended to certain singular curves, in a way that is compatible with degenerations. If X is a Deligne-Mumford stable curve (or a cuspidal curve), we describe the enumerative and local geometry of T(X); this is applied to show that some singular curves are uniquely determined by T(X). This work generalizes the preliminary steps used in AG/0008239 (with E. Sernesi) to show that a general curve of genus 3 can be recovered from its odd theta-characteristics.

math.AG↗
arXiv2000

On certain uniformity properties of curves over function fileds

Let g be an integer greater than 1. A uniform version of the Parshin-Arakelov theorem on the finiteness of the set of non-isotrivial curves of genus g over a function field, with fixed degeneracy locus, is proved. This is applied to obtain a uniform version of Manin theorem (the Mordell conjecture on rational points for curves of genus g over function fields). By a "function field" is meant the function field of a complex variety V of any dimension. If V is a curve, the uniform bounds will depend on g, on the genus of V and on the cardinality of the degeneracy locus.

math.AG↗