On the degree of irrationality of low genus $K3$ surfaces
Federico Moretti, Andrés Rojas
TL;DR
The paper studies the degree of irrationality for polarized K3 surfaces of genus $g\le 14$ by analyzing minimal-degree projections $S\dasharrow \mathbb{P}^2$ through the kernel-bundle framework and Bridgeland stability. It proves $\mathrm{irr}_L(S)\le 4$ for all such $K3$ surfaces, and shows that for certain odd genera ($g=7,9,11$) no degree-$3$ maps arise from the primitive linear system; it then constructs and classifies degree-$4$ maps via multiple mechanisms, including pairs $(E,\xi)$ with $E$ in suitable moduli spaces and nonreduced subschemes in $S^{[n]}$, yielding several components of the Brill–Noether locus $W^2_4(S,L)$. A key technical achievement is the isomorphism of Hilbert schemes $S^{[2]}\cong \mathcal{M}^{[2]}$ (and related higher-jump loci) obtained through Bridgeland stability and Fourier–Mukai transforms, enabling precise cohomological controls such as $h^1(E\otimes\mathcal{I}_\xi)$ to rule out degree-$3$ cases. The results leverage a synthesis of kernel-bundle techniques, derived-category methods, and the geometry of singular curves on K3s, uncovering a structured picture of how $W^2_4(S,L)$ governs irrationality in low genus and guiding future explorations to higher genus.
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 à la Lazarsfeld with derived category tools, and also make use of the rich theory of singular curves on $K3$ surfaces.