Extensions of curves with high degree with respect to the genus
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916047591833600 |
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| author | Ciliberto, Ciro Dedieu, Thomas |
| author_facet | Ciliberto, Ciro Dedieu, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_01851 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extensions of curves with high degree with respect to the genus Ciliberto, Ciro Dedieu, Thomas Algebraic Geometry 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. |
| title | Extensions of curves with high degree with respect to the genus |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2304.01851 |