Desingularization of Kontsevich’s compactification of twisted cubics in $$V_5$$

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 Plücker embedding, and rational curves in $$V_5$$ have been examined in various studies on Fano geometry. In this paper, we propose an explicit birational relation for the Kontsevich and Simpson compactifications of twisted cubic curves in $$V_5$$ . As a direct corollary, we obtain a desingularized model of Kontsevich compactification that induces the intersection cohomology group of Kontsevich’s space.