Remarks on Ruled surfaces and rank two bundles with canonical determinant and 4 sections

2014/11/15 by Abel Castorena, Castorena, Abel, Graciela Reyes-Ahumada +1
Mathematics · #14H60 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 14C20 #Secondary 14J26 #math.AG #msc:14C20 #msc:14H60 #msc:14J26

paper · pdf · doi:10.48550/arxiv.1411.4155

14 pages. Second Version. Corrected typos. Minor changes in the proof of the main Theorem. Added references

arxiv created 2015/03/25 · arxiv updated 2015/03/26

Abstract

Let C be a smooth irreducible complex projective curve of genus g and let B^k(2,K_C) be the Brill-Noether loci parametrizing classes of (semi)-stable vector bundles E of rank two with canonical determinant over C with h⁰(C,E)≥ k. We show that B⁴(2,K_C) it has an irreducible component \mathcal B of dimension 3g-13 on a general curve C of genus g≥ 8. Moreover, we show that for the general element [E] of \mathcal B, E fits in a exact sequence 0→\mathcal O_C(D)→ E→ K_C(-D)→ 0, with D a general effective divisor of degree three, and the corresponding coboundary map ∂: H⁰(C,K_C(-D))→ H¹(C,\mathcal O_C(D)) has cokernel of dimension three.

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