Equivariant enumerative geometry

TL;DR

This paper develops an equivariant conservation of number principle to compute generalized Euler numbers in the presence of group actions, enabling new results in equivariant enumerative geometry, exemplified by counting lines on symmetric cubic surfaces.

Contribution

It introduces an equivariant conservation of number and applies it to enumerate lines on symmetric cubic surfaces under group actions, advancing equivariant enumerative geometry.

Findings

Proves that symmetric cubic surfaces have lines with specific orbit types under group action.

Shows real symmetric cubic surfaces can only have 3 or 27 real lines.

Develops a framework for counting geometric objects with symmetry using local indices.

Abstract

We formulate an equivariant conservation of number, which proves that a generalized Euler number of a complex equivariant vector bundle can be computed as a sum of local indices of an arbitrary section. This involves an expansion of the Pontryagin--Thom transfer in the equivariant setting. We leverage this result to commence a study of enumerative geometry in the presence of a group action. As an illustration of the power of this machinery, we prove that any smooth complex cubic surface defined by a symmetric polynomial has 27 lines whose orbit types under the $S_4$-action on $\mathbb{C}P^3$ are given by $[S_4/C_2]+[S_4/C_2'] + [S_4/D_8]$, where $C_2$ and $C_2'$ denote two non-conjugate cyclic subgroups of order two. As a consequence we demonstrate that a real symmetric cubic surface can only contain 3 or 27 real lines.