The flex divisor of a K3 surface

TL;DR

This paper investigates the properties of the flex divisor on primitively polarized K3 surfaces, establishing its linear system placement and extending its definition over the entire moduli space.

Contribution

It introduces a precise description of the flex divisor's linear system and generalizes its concept across the moduli space of polarized K3 surfaces.

Findings

Flex divisor lies in the linear system |n_dL| with n_d=(2d+1)C(d)^2.

Defined a consistent notion of flex divisor over the entire moduli space.

Connected the flex divisor's properties to the geometry of K3 surfaces.

Abstract

The flex divisor of a primitively polarized K3 surface $(X,L)$ of degree $L^2=2d$ is, generically, the locus of all points $x\in X$ for which there exists a pencil $V\subset |L|$ whose base locus is $\{x\}$. We show that the flex divisor lies in the linear system $|n_dL|$ where $n_d=(2d+1)C(d)^2$ and $C(d)$ is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space $F_{2d}$ of polarized K3 surfaces.