Characterizing cubic hypersurfaces via projective geometry

TL;DR

This paper uses algebraic K-theory relations to characterize high-dimensional cubic hypersurfaces and related varieties, extending previous conditions and identifying unique geometric signatures.

Contribution

It introduces a new approach using the cut and paste relation in K_0(Var_k) to characterize cubic hypersurfaces and related complete intersections under certain conditions.

Findings

Characterizes cubic hypersurfaces via projective geometry and K-theory relations.

Extends characterization to complete intersections of quadrics and quartics.

Identifies specific cut and paste relations unique to cubic hypersurfaces.

Abstract

We use the cut and paste relation $[Y^{[2]}] = [Y][\mathbb{P}^m] + \mathbb{L}^2 [F(Y)]$ in $K_0(\text{Var}_k)$ of Galkin--Shinder for cubic hypersurfaces arising from projective geometry to characterize cubic hypersurfaces of sufficiently high dimension under certain numerical or genericity conditions. Removing the conditions involving the middle Betti number from the numerical conditions used extends the possible $Y$ to cubic hypersurfaces, complete intersections of two quadric hypersurfaces, or complete intersections of two quartic hypersurfaces. The same method also gives a family of other cut and paste relations that can only possibly be satisfied by cubic hypersurfaces.