We study the K-moduli space of Fano threefolds obtained by first blowing up ℙ^3 along an elliptic quartic curve C and then blowing up a fiber ℓ_p of the exceptional divisor E → C. We prove that this K-moduli space is isomorphic to a VGIT quotient parametrizing pairs (C,p), with linearization induced by the CM line bundle. In particular, we classify all K-(semi/poly)stable members of this deformation family. The main new ingredients in the proof include the geometry of the Hilbert scheme of elliptic quartic curves, deformation theory of Fano–K3 pairs, and optimal bounds on the local volumes of threefold singularities.

We complete the classification of regular generically free actions of finite groups on del Pezzo surfaces, up to birational equivalence. As a byproduct, we settle several open problems in equivariant birational geometry, e.g., we classify birationally rigid actions on del Pezzo surfaces.

We study linearizability of actions of finite groups on cubic threefolds with non-isolated singularities.

We study the K-stability of singular Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system is base-point-free but not very ample.

We prove that if X is a smooth Fano threefold and L is an ample ℚ-divisor such that (X,L) is K-polystable, then the automorphism group Aut(X) is reductive. This verifies the reductivity statement predicted by the Yau–Tian–Donaldson conjecture in the setting of smooth Fano threefolds with arbitrary ample polarisation.

A variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We show that a smooth Fano 3-fold not satisfying Condition (A) is K-polystable unless it is contained in eight exceptional deformation families (seven of them consist of one smooth member, and one of them has two-parameter moduli).

We study unirationality of actions of finite groups on Fano threefolds.

We classify pairs (X,G) consisting of a (possibly singular) cubic threefold X⊂ℙ^4 and a finite subgroup G⊂Aut(X) such that X is G-birationally rigid, i.e., X is a G-Mori fiber space (over a point), and X is not G-birational to any G-Mori fibre space that is not G-biregular to X.

We find all K-polystable limits of smooth Fano threefolds in family 3.10.

We construct explicit unirational del Pezzo surfaces of degree 1 with arithmetic Picard rank one over ℚ, 𝔽_5, and ℂ(t). Moreover, we prove that every smooth real geometrically rational surface is unirational over ℝ if and only if it has a real point.

In this paper, we study finite subgroups G⊂Aut(ℙ^n) such that ℙ^n is G-birationally rigid. For each n⩾ 3, we prove that Aut(ℙ^n) contains at most finitely many such subgroups up to conjugation. For n=4, we prove that ℙ^4 is G-birationally superrigid if G≃PSp_4(𝐅_3).

We study rationality properties of real singular cubic threefolds.

A smooth variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We classify smooth Fano 3-folds that satisfy Condition (A).

We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds.

We study linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program.

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 $\mathfrak{S}_4$ into the space Cremona group.

We study K-stability of smooth Fano threefolds of Picard rank 2 and degree 22 which can be obtained by blowing up a smooth complete intersection of two quadrics in ℙ^5 along a conic. We also describe the automorphism groups of these threefolds.