Grothendieck ring of pairs of quasi-projective varieties

TL;DR

This paper introduces a Grothendieck ring for pairs of complex quasi-projective varieties, explores its algebraic structures, and verifies a conjecture relating symmetric powers of orbifold points to pairs of varieties.

Contribution

It defines a new Grothendieck ring for pairs of varieties, describes its algebraic structures, and confirms a conjecture about symmetric powers of orbifold points.

Findings

Defined a Grothendieck ring for pairs of varieties

Established $mbda$-structures and power structures on the ring

Confirmed the conjecture relating symmetric powers of orbifold points

Abstract

We define a Grothendieck ring of pairs of complex quasi-projective varieties (that is a variety and a subvariety). We describe $\lambda$-structures and a power structure on/over this ring. We show that the conjectual symmetric power of the projective line with several orbifold points described by A.Fonarev is consistent with the symmetric power of this line with points as a pair of varieties.