Rational quartic curves in the Mukai-Umemura variety

초록

Let X be a Fano threefold of index one and degree 22 with Pic(X) congruent to Z. Such a threefold X can be realized as the zero locus of a regular section s of (boolean AND(2) U*)(circle plus 3) over the Grassmannian Gr(3, V ), where dim V = 7 and is the universal subbundle. When the section s is given by the net of the SL2-invariant skew-symmetric forms, we call it 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 we compute its Poincar & eacute; polynomial by applying Bialynicki-Birula's theorem. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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