Ivan Cheltsov

Sextic Double Solids

We study properties of double covers of ℙ3 ramified along nodal sextic surfaces such as nonrationality, ℚ-factoriality, potential density, and elliptic fibration structures. We also consider some relevant problems over fields of positive characteristic.

On factoriality of nodal threefolds

We prove the Q-factoriality of a nodal hypersurface in P4 of degree n with at most (n−1) 2 4 nodes and the Q-factoriality of a double cover of P3 branched over a nodal surface of degree 2r with at most (2r−1)r 3 nodes.

Log canonical thresholds of certain Fano hypersurfaces

We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an important application, we show that they have Kähler–Einstein metrics if they are general.

Affine cones over smooth cubic surfaces

We show that affine cones over smooth cubic surfaces do not admit non-trivial

On exceptional quotient singularities

\mathbb{G}_a

Weighted Fano threefold hypersurfaces

-actions.

Five embeddings of one simple group

We study exceptional quotient singularities. In particular, we prove an exceptionality criterion in terms of the ‐invariant of Tian, and utilize it to classify four-dimensional and five-dimensional exceptional quotient singularities.

Birationally rigid Fano threefold hypersurfaces

Abstract We study birational transformations into elliptic fibrations and birational automorphisms of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces with terminal singularities classified by A. R. Iano-Fletcher, J. Johnson, J. Kollár, and M. Reid.

Cremona groups and the icosahedron

We propose a new method to study birational maps between Fano varieties based on multiplier ideal sheaves. Using this method, we prove equivariant birational rigidity of four Fano threefolds acted on by the group A6. As an application, we obtain that Bir(P 3 ) has at least five non-conjugate subgroups isomorphic to A6.

Factorial threefold hypersurfaces

We prove that every quasi-smooth hypersurface in the 95 families of weighted Fano threefold hypersurfaces is birationally rigid.