Extensions of curves with high degree with respect to the genus

2023/04/04 by Ciro Ciliberto, Thomas Dedieu · 1 citation
Mathematics · Arts and Humanities · #math.AG

paper · pdf · doi:10.46298/epiga.2024.11202

Abstract

We classify linearly normal surfaces S ⊂ P^r+1 of degree d such that 4g-4 ≤ d ≤ 4g+4, where g>1 is the sectional genus (it is a classical result that for larger d there are only cones). We apply this to the study of the extension theory of pluricanonical curves and genus 3 curves, whenever they verify Property N₂, using and slightly expanding the theory of integration of ribbons of the authors and E.~Sernesi. We compute the corank of the relevant Gaussian maps, and we show that all ribbons over such curves are integrable, and thus there exists a universal extension. We carry out a similar program for linearly normal hyperelliptic curves of degree d≥ 2g+3. We classify surfaces having such a curve C as a hyperplane section, compute the corank of the relevant Gaussian maps, and prove that all ribbons over C are integrable if and only if d=2g+3. In the latter case we obtain the existence of a universal extension.

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