276 research outputs found
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.Comment: 36 pages. Final version, to appear in JEM
Rank of divisors on graphs: an algebro-geometric analysis
The divisor theory for graphs is compared to the theory of linear series on
curves through the correspondence associating a curve to its dual graph. An
algebro-geometric interpretation of the combinatorial rank is proposed, and
proved in some cases.Comment: Final version incorporating referee remarks. Dedicated to Joe Harris
for his sixtiest birthday. 23 page
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.Comment: Minor revisions. New numbering matches the journal version. 41 page
Algebraic and tropical curves: comparing their moduli spaces
We construct the moduli space for equivalence classes of n-pointed tropical
curves of genus g, together with its compactification given by weighted
tropical curves, and establish some of its basic topological properties. We
compare it to the moduli spaces of smooth and stable algebraic curves, from the
combinatorial, the topological, and the Teichm\"uller point of view. The paper
is written in an expository style, and it generalizes some results contained in
sections 4-6 of arXiv:1001.2815v3.Comment: Changes in numbering to match the published versio
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