Algebraic cycles and intersections of a quadric and a cubic

TL;DR

This paper proves that certain algebraic varieties formed by intersecting a quadric and a cubic have a special decomposition in their Chow rings, revealing K3-like properties and extending to nodal cubic hypersurfaces.

Contribution

It establishes a multiplicative Chow-K"unneth decomposition for these intersections and for resolutions of singularities of nodal cubic hypersurfaces, advancing understanding of their Chow rings.

Findings

Chow ring of the intersection exhibits K3-like behavior.

Existence of multiplicative Chow-K"unneth decomposition for these varieties.

Extension of results to resolutions of nodal cubic hypersurfaces.

Abstract

Let $Y$ be a smooth complete intersection of a quadric and a cubic in $\mathbb{P}^n$, with $n$ even. We show that $Y$ has a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of (powers of) $Y$ displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow-K\"unneth decomposition for the resolution of singularities of a general nodal cubic hypersurface of even dimension.