Rational quartic curves in the Mukai-Umemura variety

Kiryong Chung; Jaehyun Kim; Jeong-Seop Kim

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arXiv · 2412.17721

Rational quartic curves in the Mukai-Umemura variety

Kiryong Chung·Jaehyun Kim·Jeong-Seop Kim

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Abstract

Let $X$ be the Fano threefold of index one, degree $22$, and $\mathrm{Pic}(X)\cong\mathbb{Z}$. Such a threefold $X$ can be realized by a regular zero section $\mathbf{s}$ of $(\bigwedge^2\mathcal{F}^{*})^{\oplus 3}$ over Grassmannian variety $\mathrm{Gr}(3,V)$, $\dim V=7$ with the universal subbundle $\mathcal{F}$. When the section $\mathbf{s}$ is given by the net of the $\mathrm{SL}_2$-invariant skew forms, we call it by the Mukai-Umemura (MU) variety. In this paper, we prove that the Hilbert scheme of rational quartic curves in the MU-variety is smooth and compute its Poincar\'e polynomial by applying the Bia{\l}ynicki-Birula's theorem.

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BibTeX RIS

Kiryong Chung, Jaehyun Kim, Jeong-Seop Kim. 2024-12-23. Rational quartic curves in the Mukai-Umemura variety. https://arxiv.org/abs/2412.17721

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