Margarida Melo

On the tropical Torelli map

Abstract We construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map.

Fourier–Mukai and autoduality for compactified Jacobians. I

Abstract To every singular reduced projective curve X one can associate, following Esteves, many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We prove that, for a reduced curve with locally planar singularities, the integral (or Fourier–Mukai) transform with kernel the Poincaré sheaf from the derived category of the generalized Jacobian of X to the derived category of any fine compactified Jacobian of X is fully faithful, generalizing a previous result of Arinkin in the case of integral curves. As a consequence, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the identity of the Picard scheme of any fine compactified Jacobian of X and that algebraic equivalence and numerical equivalence of line bundles coincide on any fine compactified Jacobian, generalizing previous results of Arinkin, Esteves, Gagné, Kleiman, Rocha, and Sawon. The paper contains an Appendix in which we explain how our work can be interpreted in view of the Langlands duality for the Higgs bundles as proposed by Donagi–Pantev.

Comparing perfect and 2nd Voronoi decompositions: the matroidal locus

We compare two rational polyhedral admissible decompositions of the cone of positive definite quadratic forms: the perfect cone decomposition and the 2nd Voronoi decomposition. We determine which cones belong to both the decompositions, thus providing a positive answer to a conjecture of Alexeev and Brunyate (Invent. Math. doi:10.1007/s00222-011-0347-2, 2011). As an application, we compare the two associated toroidal compactifications of the moduli space of principal polarized abelian varieties: the perfect cone compactification and the 2nd Voronoi compactification.

Fine compactified Jacobians

eron model of the Jacobian of the general fiber, and thus it provides a modular compactification of it. We show that each fine compactified Jacobian of X admits a stratification in terms of certain fine compactified Jacobians of partial normalizations of X and, moreover, that it can be realized as a quotient of the smooth locus of a suitable fine compactified Jacobian of the total blowup of X .F inally, we determine when a fine compactified Jacobian is isomorphic to the corresponding Oda-Seshadri’s coarse compactified Jacobian.

Geometric invariant theory for polarized curves

Introduction.- Singular Curves.- Combinatorial Results.- Preliminaries on GIT.- Potential Pseudo-stability Theorem.- Stabilizer Subgroups.- Behavior at the Extremes of the Basic Inequality.- A Criterion of Stability for Tails.- Elliptic Tails and Tacnodes with a Line.- A Strati_cation of the Semistable Locus.- Semistable, Polystable and Stable Points (part I).- Stability of Elliptic Tails.- Semistable, Polystable and Stable Points (part II).- Geometric Properties of the GIT Quotient.- Extra Components of the GIT Quotient.- Compacti_cations of the Universal Jacobian.- Appendix: Positivity Properties of Balanced Line Bundles.

Potential Pseudo-Stability Theorem

The aim of this chapter is to generalize the Potential stability theorem (see Fact 4.22) for smaller values of d. The main result is the following theorem, which we call Potential pseudo-stability Theorem because of its relations with the pseudo-stable curves (see Definition 2.1(ii)).

Appendix: Positivity Properties of Balanced Line Bundles

The aim of this Appendix is to investigate positivity properties of balanced line bundles of sufficiently high degree on (reduced) Gorenstein curves. The results obtained here are applied in this manuscript only for quasi-wp-stable curves; however we decided to present these results in the Gorenstein case for two reasons: firstly, we think that these results are interesting in their own (in particular we will generalize our proofs extend without any modifications to the Gorenstein case.

Extra Components of the GIT Quotient

So far, we have considered the action of \(\mathrm{GL}_{r+1}\) over Hilb d , and we have restricted our attention to \(\mathrm{Ch}^{-1}(\mathrm{Chow}_{d}^{\mathit{ss}})^{o}\) and \(\mathrm{Hilb}_{d}^{\mathit{ss},o}\), the Chow or Hilbert semistable loci consisting of connected curves. It is very natural to ask if there are Chow or Hilbert semistable points \([X \subset \mathbb{P}^{r}] \in \mathrm{ Hilb}_{d}\) with X not connected. In this chapter we will answer this question.

Behavior at the Extremes of the Basic Inequality

Recall from Corollary 5.6(i) that if \([X \subset \mathbb{P}^{r}] \in \mathrm{Hilb}_{d}\) is Chow semistable with X connected and d > 2(2g − 2), then X is quasi-wp-stable and \(\mathcal{O}_{X}(1)\) is properly balanced.

Stability of Elliptic Tails

In this chapter, we will use the criterion of stability for tails (Proposition 8.3) in order to study the stability of elliptic curves for \(\frac{7} {2}(2g - 2) < d \leq 4(2g - 2)\). We notice that in this range—by the basic inequality ( 3.1)—it suffices to consider the elliptic curves of degree 4.