On the variety of tritangential planes to a general K3 surface of degree 6 and genus 4 in $\mP^4$

TL;DR

This paper investigates the geometry of tritangential planes to a general K3 surface of degree 6 and genus 4 in projective 4-space, revealing their dimension, irreducibility, smoothness, and degree within the Grassmannian.

Contribution

It provides a detailed geometric analysis of the variety of tritangential planes to a general K3 surface, including its dimension, smoothness, and degree, linking it to the Beauville involution.

Findings

The variety of tritangential planes has dimension 3.

It is irreducible and mostly smooth, with 210 points of at most multiplicity 2.

The degree of this variety in the Grassmannian is 152.

Abstract

Let $S\subset \mP^4$ be a general K3 surface of degree 6 and genus 4. In this paper we study the irreducible variety $X_S$ of \emph{tritangential planes} to $S$ whose general point is a plane that intersects $S$ in a curvilinear scheme of length six supported at three non collinear points. The variety $X_S$ can be identified as the relevant part of the fixed locus of the so called \emph{Beauville involution} defined on the Hilbert scheme $S[3]$ of 0--dimensional schemes of length three of $S$. In this paper we prove that: (a) $X_S$ has dimension 3, is irreducible and smooth, except for 210 points that are at most of multiplicity 2 for $X_S$; (b) $X_S$, in its natural embedding in the Grassmannian $\mathbb G(2,4)\subset \mP^9$ of planes in $\mP^4$, has degree $152$.