# A desingularization of Kontsevich's compactification of twisted cubics in $V_5$

## TL;DR

This paper constructs a desingularized model of Kontsevich's compactification of twisted cubic curves in the Fano threefold V_5, providing explicit birational relations and insights into the intersection cohomology of the space.

## Contribution

It introduces an explicit birational relation between Kontsevich and Simpson compactifications of twisted cubics in V_5, leading to a desingularized model of Kontsevich's space.

## Key findings

- Desingularized model of Kontsevich's compactification constructed.
- Explicit birational relation between Kontsevich and Simpson compactifications.
- Intersection cohomology of Kontsevich's space analyzed.

## Abstract

By definition, the del Pezzo $3$-fold $V_5$ is the intersection of $\mathrm{Gr}(2,5)$ with three hyperplanes in $\mathbb{P}^9$ under the Pl\"ucker embedding. Rational curves in $V_5$ have been studied in various contents of Fano geometry. In this paper, we propose an explicit birational relation of the Kontsevich and Simpson compactifications of twisted cubic curves in $V_5$. As a direct corollary, we obtain a desingularized model of Kontsevich compactification which induces the intersection cohomology group of Kontsevich's space.