Singularities of elliptic curves in $K3$ surfaces and the Beauville-Voisin zero-cycle

TL;DR

This paper provides a geometric proof that the second Chern class of a primitively polarized K3 surface equals 24 times its Beauville-Voisin zero-cycle, under certain singularity hypotheses on elliptic curve families.

Contribution

It offers a new geometric proof of a key property of K3 surfaces relating their second Chern class to the Beauville-Voisin cycle, assuming specific singularity conditions.

Findings

Second Chern class equals 24 times the Beauville-Voisin cycle in Chow group

Geometric proof under hypotheses on elliptic curve singularities

Supports conjectures for very general polarized K3 surfaces

Abstract

Under some hypotheses on the singular type of the one-parameter family of elliptic curves in a primitively polarized $K3$ surface $S$ determined by its polarization (which is expected to be true for a very general polarized $K3$ surface), we give a more geometric proof of the fact that the second Chern class of $S$ is equal to $24 \cdot o_S$ in the Chow group of $0$-cycles where $o_S$ is the Beauville-Voisin canonical $0$-cycle.