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ON THE DEGREE OF IRRATIONALITY OF LOW GENUS
$K3$ SURFACES
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu , First View
- Published online by Cambridge University Press:
- 19 March 2025, pp. 1-36
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- Article
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Given a general polarized
$K3$ surface 
$S\subset \mathbb P^g$ of genus 
$g\le 14$, we study projections 
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 
of degree 
$3$ induced by the primitive linear system. Our methods combine vector bundle techniques à la Lazarsfeld with derived category tools and also make use of the rich theory of singular curves on 
$K3$ surfaces.
of minimal degree and their variational structure. In particular, we prove that the degree of irrationality of all such surfaces is at most 
of degree 
