Results 1  10
of
18
Linear series on metrized complexes of algebraic curves
, 2014
"... A metrized complex of algebraic curves over an algebraically closed field κ is, roughly speaking, a finite metric graph Γ together with a collection of marked complete nonsingular algebraic curves Cv over κ, one for each vertex v of Γ; the marked points on Cv are in bijection with the edges of Γ i ..."
Abstract

Cited by 18 (4 self)
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A metrized complex of algebraic curves over an algebraically closed field κ is, roughly speaking, a finite metric graph Γ together with a collection of marked complete nonsingular algebraic curves Cv over κ, one for each vertex v of Γ; the marked points on Cv are in bijection with the edges of Γ incident to v. We define linear equivalence of divisors and establish a RiemannRoch theorem for metrized complexes of curves which combines the classical RiemannRoch theorem over κ with its graphtheoretic and tropical analogues from [AC, BN, GK, MZ], providing a common generalization of all of these results. For a complete nonsingular curve X defined over a nonArchimedean field K, together with a strongly semistable model X for X over the valuation ring R of K, we define a corresponding metrized complex CX of curves over the residue field κ of K and a canonical specialization map τCX ∗ from divisors on X to divisors on CX which preserves degrees and linear equivalence. We then establish generalizations of the specialization lemma from [B] and its weighted graph analogue from [AC], showing that the rank of a divisor cannot go down under specialization from X to CX. As an application, we establish a concrete link between specialization of divisors from curves to metrized complexes and the EisenbudHarris theory [EH] of limit linear series. Using this link, we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove, among other things, that any degeneration of a grd in a regular family of semistable curves is a limit grd on the special fiber.
Linear series on semistable curves.
, 2008
"... Abstract. For a semistable curve X of genus g, the number h 0 (X, L) is studied for line bundles L of degree d parametrized by the compactified Picard scheme. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: X has two components; X is any semi ..."
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Cited by 9 (2 self)
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Abstract. For a semistable curve X of genus g, the number h 0 (X, L) is studied for line bundles L of degree d parametrized by the compactified Picard scheme. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: X has two components; X is any semistable curve and d = 0 or d = 2g − 2; X is stable, free from separating nodes, and d ≤ 4. These results are shown to be sharp. Applications to the Clifford index, to the combinatorial description of hyperelliptic curves, and to plane quintics are given. Contents
The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces
, 2009
"... We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define ..."
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Cited by 4 (0 self)
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We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define local coordinates and construct foliations on the moduli spaces. We use these constructions to give a new proof of the Diaz’ bound on the dimension of complete subvarieties of the moduli spaces. Geometrically, we study the properties of meromorphic differentials with real periods, and their degenerations.
Naturality of Abel maps
"... Abstract. We give a combinatorial characterization of nodal curves admitting a natural (i.e. compatible with and independent of specialization) dth Abel map for any d ≥ 1. ..."
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Cited by 3 (1 self)
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Abstract. We give a combinatorial characterization of nodal curves admitting a natural (i.e. compatible with and independent of specialization) dth Abel map for any d ≥ 1.