Double Veronese cones with 28 nodes

Hamid Abban; Ivan Cheltsov; Jihun Park; Constantin Shramov

doi:10.4171/lem/1067

JournalslemOnline First

Double Veronese cones with 28 nodes

Hamid Abban
University of Nottingham, Nottingham, UK
Ivan Cheltsov
The University of Edinburgh, Edinburgh, UK; National Research University Higher School of Economics, Moscow, Russia
Jihun Park
Institute for Basic Science, Gyeongbuk, Korea; Pohang University of Science and Technology, Gyeongbuk, Korea
Constantin Shramov
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia; National Research University Higher School of Economics, Moscow, Russia

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Abstract

We study nodal del Pezzo $3$ -folds of degree $1$ (also known as double Veronese cones) with $28$ singularities, which is the maximal possible number of singularities for such varieties. We show that they are in one-to-one correspondence with smooth plane quartics and use this correspondence to study their automorphism groups. As an application, we find all $G$ -birationally rigid varieties of this kind, and construct an infinite number of non-conjugate embeddings of the group $S_{4}$ into the space Cremona group.

Cite this article

Hamid Abban, Ivan Cheltsov, Jihun Park, Constantin Shramov, Double Veronese cones with 28 nodes. Enseign. Math. (2024), published online first

DOI 10.4171/LEM/1067