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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\"uckerembedding. Rational curves in $V_5$ have been studied in various contents ofFano geometry. In this paper, we propose an explicit birational relation of theKontsevich and Simpson compactifications of twisted cubic curves in $V_5$. As adirect corollary, we obtain a desingularized model of Kontsevichcompactification which induces the intersection cohomology group ofKontsevich's space.

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