Lucia Caporaso

Uniformity of rational points

1. Uniformity and correlation 1 1.

TORELLI THEOREM FOR GRAPHS AND TROPICAL CURVES

Algebraic curves have a discrete analog 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. By contrast, we prove that, in a suitably defined sense, the tropical Torelli map has degree one. Finally, we describe some natural posets associated to a graph and prove that they characterize its Delaunay decomposition.

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.

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.

Parameter spaces for curves on surfaces and enumeration of rational curves

Let S be a smooth, minimal rational surface. The geometry of the Severi variety parametrising irreducible, rational curves in a given linear system on S is studied. The results obtained are applied to enumerative geometry, in combination with ideas from Quantum Cohomology. Formulas enumerating rational curves are found, some of which generalised Kontsevichs formula for plane curves.

Gonality of algebraic curves and graphs

We define d-gonal weighted graphs using “harmonic indexed” morphisms, and prove that a combinatorial locus of (overline{M_{g}}) contains a d-gonal curve if the corresponding graph is d-gonal and of Hurwitz type. Conversely the dual graph of a d-gonal stable curve is equivalent to a d-gonal graph of Hurwitz type. The hyperelliptic case is studied in detail. For r ≥ 1, we show that the dual graph of a (d, r)-gonal stable is the underlying graph of a tropical curve admitting a degree-d divisor of rank at least r.

Geometry of tropical moduli spaces and linkage of graphs

We prove the following linkage theorem: two p-regular graphs of the same genus can be obtained from one another by a finite alternating sequence of one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the linkage theorem to prove that various moduli spaces of tropical curves are connected through codimension one.

Geometry of the theta divisor of a compactified jacobian

The object of this paper is the theta divisor of the compactified jacobian of a nodal curve. We determine its irreducible components and give it a geometric interpretation. A characterization of hyperelliptic irreducible stable curves is appended as an application.

How Many Rational Points Can a Curve Have

This paper is concerned with two conjectures in number theory describing the behavior of the number of rational points on an algebraic curve defined over a number field, as that curve varies.

Combinatorial Properties of Stable Spin Curves

Abstract 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. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.