Extremal higher codimension cycles of the space of complete conics

TL;DR

This paper computes the effective and nef cones of higher codimension cycles in the space of complete conics and analyzes their structure using Białynicki-Birula decomposition.

Contribution

It provides explicit descriptions of the cones of effective and nef cycles of all codimensions in the space of complete conics, and relates these to the Białynicki-Birula cell decomposition.

Findings

Explicit cones for all codimension cycles in the space of complete conics.

Comparison between cell decomposition cones and effective/nef cones.

Insights into the geometric structure of the space of complete conics.

Abstract

Let $M$ denote the space of complete conics. We compute the cone of effective and numerically effective $k$-cycles of $M$, $\mathrm{Eff}_k(M)$ and $\mathrm{Nef}_k(M)$, respectively. In addition, we compute the Bia\l{}ynicki-Birula cell-decomposition of $M$ with respect to a $\mathbb{C}^*$-action and compare the cone generated by the closure of these cells to the cones $\mathrm{Eff}_k(M)$ and $\mathrm{Nef}_k(M)$.