arXiv subjects

Margarida Melo

Publications and source records attributed to Margarida Melo.

24 records · Page 2Linked to original sources

arXiv2011

Fine compactified Jacobians

We study Esteves's fine compactified Jacobians for nodal curves. We give a proof of the fact that, for a one-parameter regular local smoothing of a nodal curve $X$, the relative smooth locus of a relative fine compactified Jacobian is isomorphic to the Néron 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 $X$. Finally, we determine when a fine compactified Jacobian is isomorphic to the corresponding Oda-Seshadri's coarse compactified Jacobian.

math.AG↗

arXiv2010

On the tropical Torelli map

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.

math.AG↗

arXiv2008

Compactified Picard stacks over the moduli stack of stable curves with marked points

In this paper we give a construction of algebraic (Artin) stacks endowed with a modular map onto the moduli stack of n-pointed stable curves of genus g, for g greater than 2. These stacks are smooth, irreducible and have dimension 4g-3+n, yielding a geometrically meaningful compactification of the degree d universal Picard stack over the moduli stack of smooth curves with marked points.

math.AG↗

arXiv2008

Compactified Picard stacks over $\bar{\mathcal M}_g$

We study algebraic (Artin) stacks over $\bar{\mathcal M}_g$ giving a functorial way of compactifying the relative degree $d$ Picard variety for families of stable curves. We also describe for every $d$ the locus of genus $g$ stable curves over which we get Deligne-Mumford stacks strongly representable over $\bar{\mathcal M}_g$.

math.AG↗

arXiv2008

On the complexity group of stable curves

In this paper, we study combinatorial properties of stable curves. To the dual graph of any nodal curve, it is naturally associated a group, which is the group of components of the Néron model of the generalized Jacobian of the curve. We study the order of this group, called the complexity. In particular, we provide a partial characterization of the stable curves having maximal complexity, and we provide an upper bound, depending only on the genus $g$ of the curve, on the maximal complexity of stable curves; this bound is asymptotically sharp for $g\gg 0$. Eventually, we state some conjectures on the behavior of stable curves with maximal complexity, and prove partial results in this direction.

math.AG↗

arXiv2006

Combinatorial aspects of nodal curves

To any nodal curve $C$ is associated the degree class group, a combinatorial invariant which plays an important role in the compactification of the generalised Jacobian of $C$ and in the construction of the Néron model of the Picard variety of families of curves having $C$ as special fibre. In this paper we study this invariant. More precisely, we construct a wide family of graphs having cyclic degree class group and we provide a recursive formula for the cardinality of the degree class group of the members of this family. Moreover, we analyse the behaviour of the degree class group under standard geometrical operations on the curve, such as the blow up and the normalisation of a node.

math.AG↗