Seshadri constants of $K3$ surfaces of degrees 6 and 8

TL;DR

This paper calculates Seshadri constants for certain K3 surfaces and characterizes when these constants lie between 1 and 2 based on the generation of the surface's ideal.

Contribution

It provides explicit Seshadri constant values for K3 surfaces of degrees 6 and 8 and characterizes the constants for higher degrees in relation to the ideal's generators.

Findings

Seshadri constants for degree 6 and 8 K3 surfaces are computed.

For higher degrees, the constants are between 1 and 2 if and only if the ideal is not generated solely by quadrics.

When the ideal is not generated by quadrics, the Seshadri constant equals 3/2.

Abstract

We compute Seshadri constants $\eps(X):= \eps(\O_X(1))$ on $K3$ surfaces $X$ of degrees 6 and 8. Moreover, more generally, we prove that if $X$ is any embedded $K3$ surface of degree $2r-2 \geq 8$ in $\PP^r$ not containing lines, then $1 < \eps(X) <2$ if and only if the homogeneous ideal of $X$ is not generated by only quadrics (in which case $\eps(X)=3/2$).