Ciro Ciliberto ; Thomas Dedieu - Extensions of curves with high degree with respect to the genus

epiga:11202 - Épijournal de Géométrie Algébrique, July 9, 2024, Special volume in honour of Claire Voisin - https://doi.org/10.46298/epiga.2024.11202
Extensions of curves with high degree with respect to the genusArticle

Authors: Ciro Ciliberto ; Thomas Dedieu

    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.


    Volume: Special volume in honour of Claire Voisin
    Published on: July 9, 2024
    Accepted on: January 27, 2024
    Submitted on: April 17, 2023
    Keywords: Mathematics - Algebraic Geometry

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