The Singular Locus of the Theta Divisor and Quadrics through a Canonical Curve - Mathematics > Algebraic GeometryReportar como inadecuado




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Abstract: A section K on a genus g canonical curve C is identified as the key tool toprove new results on the geometry of the singular locus Theta s of the thetadivisor. The K divisor is characterized by the condition of linear dependenceof a set of quadrics containing C and naturally associated to a degree geffective divisor on C. K counts the number of intersections of specialvarieties on the Jacobian torus defined in terms of Theta s. It also identifiessections of line bundles on the moduli space of algebraic curves, closelyrelated to the Mumford isomorphism, whose zero loci characterize specialvarieties in the framework of the Andreotti-Mayer approach to the Schottkyproblem, a result which also reproduces the only previously known case g=4.This new approach, based on the combinatorics of determinantal relations fortwo-fold products of holomorphic abelian differentials, sheds light on basicstructures, and leads to the explicit expressions, in terms of theta functions,of the canonical basis of the abelian holomorphic differentials and of theconstant defining the Mumford form. Furthermore, the metric on the moduli spaceof canonical curves, induced by the Siegel metric, which is shown to beequivalent to the Kodaira-Spencer map of the square of the Bergman reproducingkernel, is explicitly expressed in terms of the Riemann period matrix only, aresult previously known for the trivial cases g=2 and g=3. Finally, the inducedSiegel volume form is expressed in terms of the Mumford form.



Autor: Marco Matone, Roberto Volpato

Fuente: https://arxiv.org/







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