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arXiv · 2401.03821

On the degree of irrationality of low genus $K3$ surfaces

Federico Moretti·Andrés Rojas
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Abstract

Given a general polarized $K3$ surface $S\subset \mathbb P^g$ of genus $g\le 14$, we study projections $S\hookrightarrow \mathbb P^g\dashrightarrow \mathbb P^2$ of minimal degree and their variational structure. In particular, we prove that the degree of irrationality of all such surfaces is at most $4$, and that for $g=7,8,9,11$ there are no rational maps $S\dashrightarrow \mathbb P^2$ of degree $3$ induced by the primitive linear system. Our methods combine vector bundle techniques \`a la Lazarsfeld with derived category tools, and also make use of the rich theory of singular curves on $K3$ surfaces.

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Federico Moretti, Andrés Rojas. 2024-01-08. On the degree of irrationality of low genus $K3$ surfaces. https://doi.org/10.1017/s1474748024000525

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