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In this paper we study 14 families among 85 families of anticanonically embedded Q-Fano threefolds weighted complete intersections of codimension 2 and show that every quasismooth member is birationally birigid, that is, it is birational to…

We analyse the local structure of the K-moduli space of Fano varieties at a toric singular K-polystable Fano 3-fold, which deforms to smooth Fano 3-folds with anticanonical volume 28 and Picard rank 4. In particular, by constructing an…

Following previous work by A. Kuznetsov, we study the Fano manifolds obtained as linear sections of the spinor tenfold in $\mathbb{P}^{15}$. Up to codimension three there are finitely many such sections, up to projective equivalence. In…

We show that prime Fano threefolds $Y$ of genus 8 have a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.

We prove the birational superrigidity and nonrationality of a hypersurface in $\mathbb{P}^{6}$ of degree 6 having at most isolated ordinary double points.

We describe the 6-dimensional compact K-moduli space of Fano threefolds in deformation family No 2.18. These Fano threefolds are double covers of $\mathbb P^1\times\mathbb P^2$ branched along smooth $(2,2)$-surfaces, and…

We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then…

For $n \geq 1$, the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\mathbf{G}(n+1, 2n+2)$, of the set of graphs of…

In a first result, we describe all finitely generated factorial algebras over an algebraically closed field of characteristic zero that come with an effective multigrading of complexity one by means of generators and relations. This enables…

We study the variety of Poisson structures and compute Poisson cohomology for two families of Fano threefolds - smooth cubic threefolds and the del Pezzo quintic threefold. Along the way we reobtain by a different method earlier results of…

A powerful tool of investigation of Fano varieties is provided by exceptional collections in their derived categories. Proving the fullness of such a collection is generally a nontrvial problem, usually solved on a case-by-case basis, with…

We study unirationality and rationality of Fano threefolds of degree 18 over nonclosed fields.

We classify projective terminalizations of quotients of Fano varieties of lines on smooth cubic fourfolds by finite groups of symplectic automorphisms of the underlying cubic. We compute the second Betti number and the fundamental group of…

This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a…

We study smooth, complex Fano 4-folds X with a rational contraction onto a 3-fold, namely a rational map X-->Y that factors as a sequence of flips X-->X' followed by a surjective morphism X'->Y with connected fibers, where Y is normal,…

We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all $S_6$-invariant three-dimensional quartics…

A Fano-Enriques threefold is a three-dimensional non-Gorenstein Fano variety of index 1 with at most canonical singularities. We study the birational geometry of Fano-Enriques threefolds with terminal cyclic quotient singularities. We…

We study the Fano surface S of the Fermat cubic threefold. We prove that S is a degree 81 abelian cover of the degree 5 del Pezzo surface and that the complement of the union of 12 disjoint elliptic curves on S is a ball quotient. The…

We study weighted Fano fourfolds of K3 type realized as hypersurfaces in weighted projective spaces. Under the additional assumption that the singular locus has dimension at most one, we prove that only finitely many such families exist. We…

We show that a Hilbert scheme of conics on a Fano fourfold double cover of $\mathbb{P}^2\times\mathbb{P}^2$ ramified along a divisor of bidegree $(2,2)$ admits a $\mathbb{P}^1$-fibration with base being a hyper-K\"{a}hler fourfold. We…