Let $C$ be a non--hyperelliptic curve of genus $g$. We recall some facts about curves endowed with a base--point--free $g^1_4$. Then we prove that if the minimal degree of a surface containing the canonical model of $C$ in $\mathbb P^{g-1}_k$ is $g$, then $7\le g\le 12$ and $C$ carries exactly one $g^1_4$. As a by--product, we deduce that if the canonical model of $C$ in $\mathbb P^{g-1}_k$ is contained in a surface of degree at most $g$, then C is either trigonal or tetragonal or isomorphic to a plane sextic.
Canonical curves on surfaces of very low degree / Casnati, Gianfranco. - In: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9939. - STAMPA. - 140:4(2012), pp. 1185-1197. [10.1090/S0002-9939-2011-10979-1]
Canonical curves on surfaces of very low degree
CASNATI, GIANFRANCO
2012
Abstract
Let $C$ be a non--hyperelliptic curve of genus $g$. We recall some facts about curves endowed with a base--point--free $g^1_4$. Then we prove that if the minimal degree of a surface containing the canonical model of $C$ in $\mathbb P^{g-1}_k$ is $g$, then $7\le g\le 12$ and $C$ carries exactly one $g^1_4$. As a by--product, we deduce that if the canonical model of $C$ in $\mathbb P^{g-1}_k$ is contained in a surface of degree at most $g$, then C is either trigonal or tetragonal or isomorphic to a plane sextic.Pubblicazioni consigliate
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https://hdl.handle.net/11583/2379817
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