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Fano-Mukai fourfolds of genus 10 as compactifications of C4
European Journal of Mathematics. 2018. Vol. 4. No. 3. P. 1197–1263.
Prokhorov Y., Zaidenberg M.
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 Vs18 acted upon by SL2(ℂ). The group Open image in new window is a semidirect product Open image in new window . Furthermore, Vs18 is a GL2(ℂ)-equivariant completion of ℂ4, and as well of GL2(ℂ). The restriction of the GL2(ℂ)-action on Vs18 to Open image in new window yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold Va18 such that the group Open image in new window is a semidirect product Open image in new window . For a Fano–Mukai fourfold V18 isomorphic neither to Vs18, nor to Va18, the group Open image in new window is a semidirect product of (𝔾m)2 and a finite cyclic group whose order is a factor of 6. Besides, we establish that the affine cone over any polarized Fano–Mukai variety V18 is flexible in codimension one, and flexible if V18=Vs18.
Keywords: Fano variety
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