Fano-Mukai fourfolds of genus 10 as compactifications of ℂ⁴

2017/06/15 by Yuri Prokhorov, Mikhail Zaidenberg, Prokhorov, Yuri +1
Mathematics · #14J45 #14J50 #14R10 #14R20 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Primary 14J35 #Secondary 14L30

paper · pdf · doi:10.48550/arxiv.1706.04926

openalex publication_date 2017/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is known that the moduli space of smooth Fano-Mukai fourfolds V₁₈ of genus 10 has dimension one. We show that any such fourfold is a completion of ℂ⁴ in two different ways. Up to isomorphism, there is a unique fourfold V₁₈^{\mathrm s} acted upon by SL₂(ℂ). The group Aut(V₁₈^{\mathrm s}) is a semidirect product GL₂(ℂ)\rtimes(ℤ/2ℤ). Furthermore, V₁₈^{\mathrm s} is a GL₂(ℂ)-equivariant completion of ℂ⁴, and as well of GL₂(ℂ). The restriction of the GL₂(ℂ)-action on V₁₈^{\mathrm s} to ℂ⁴\hookrightarrow V₁₈^{\mathrm s} yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold V₁₈^{\mathrm a} such that the group Aut(V₁₈^{\mathrm a}) is a semidirect product (\mathbb G_a×\mathbb G_m)\rtimes (ℤ/2ℤ). For a Fano-Mukai fourfold V₁₈ neither isomorphic to V₁₈^{\mathrm s}, nor to V₁₈^{\mathrm a}, one has Aut⁰ (V₁₈)≅ (\mathbb G_m)², and Aut(V₁₈) is a semidirect product of Aut⁰(V₁₈) and a finite cyclic group whose order is a factor of 6.

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