Extensions of curves with high degree with respect to the genus

TL;DR

This paper classifies certain linearly normal surfaces with specific degree-genus relations and explores their extension theory, focusing on ribbons, Gaussian maps, and universal extensions for curves of genus 3 and hyperelliptic curves.

Contribution

It extends the classification of linearly normal surfaces with degrees close to four times the genus and analyzes their extension properties using ribbon integration and Gaussian maps.

Findings

Classified surfaces with degrees satisfying 4g-4 ≤ d ≤ 4g+4.

Proved all ribbons over certain curves are integrable, leading to universal extensions.

Computed the corank of Gaussian maps for these curves.

Abstract

We classify linearly normal surfaces $S \subset \mathbf{P}^{r+1}$ of degree $d$ such that $4g-4 \leq d \leq 4g+4$, 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 $N_2$, 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 $d\geq 2g+3$. 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.