Related papers: K-trivial structures on Fano complete intersection…

We study the Fano scheme of $k$-planes contained in the hypersurface cut out by a generic sum of products of linear forms. In particular, we show that under certain hypotheses, linear subspaces of sufficiently high dimension must be…

We investigate Fano schemes of conditionally generic intersections, i.e. of hypersurfaces in projective space chosen generically up to additional conditions. Via a correspondence between generic properties of algebraic varieties and events…

Let X be a Fano threefold, and let S be a K3 surface in X . Any moduli space M of simple vector bundles on S carries a holomorphic symplectic structure. Following an idea of Tyurin, we show that in some cases, those vector bundles which…

Let $X$ be a Fano manifold which is the zero scheme of a general global section $s$ in an irreducible homogenous vector bundle over a Grassmannian. We prove that the restriction of the Pl\"ucker embedding embeds $X$ projectively normal, and…

We study the geometry of the Fano schemes $\mathrm{\textbf{F}}_{k}(\mathrm{SD}_n^r)$ of the projective variety $\mathrm{SD}_n^r$ defined by the $r\times r$ minors of a symmetric $n\times n$ matrix filled with indeterminates. These schemes…

Given a generic $K3$ surface $Y_k$ of the Ap\'ery-Fermi pencil, we use the Kneser-Nishiyama technique to determine all its non isomorphic elliptic fibrations. These computations lead to determine those fibrations with 2-torsion sections T.…

Extending previous results, we prove that for $n \ge 5$ all hypersurfaces of degree $n+1$ in ${\mathbb P}^{n+1}$ with isolated ordinary double points are birational superrigid and K-stable, hence admit a weak K\"ahler--Einstein metric.

We describe in terms of spherical twists the Serre functors of many interesting semiorthogonal components, called residual categories, of the derived categories of projective varieties. In particular, we show the residual categories of Fano…

We investigate $k$-nets with $k\geq 4$ embedded in the projective plane $PG(2,\mathbb{K})$ defined over a field $\mathbb{K}$; they are line configurations in $PG(2,\mathbb{K})$ consisting of $k$ pairwise disjoint line-sets, called…

We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable Q-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano…

There is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces. We investigate symplectic aspects of this relationship, with a view to applications in low-dimensional…

A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient…

We prove birational superrigidity of every hypersurface of degree N in P^N with singular locus of dimension s, under the assumption that N is at least 2s+8 and it has only quadratic singularities of rank at least N-s. Combined with the…

Ballico proved that a smooth projective variety $X$ of degree $d$ over a finite field of $q$ elements admits a smooth hyperplane section if $q\geq d(d-1)^{\dim X}$. In this paper, we refine this criterion for higher codimensional linear…

We consider $(1,1)$-surfaces, namely, minimal compact complex surfaces $S$ with $p_g (S) =K_S^2=1$: for these the bicanonical map is a covering of degree $4$ of the plane $\mathbb{P}^2$. And we answer a question posed by Meng Chen, whether…

It is pretty well-known that toric Fano varieties of dimension k with terminal singularities correspond to convex lattice polytopes P in R^k of positive finite volume, such that intersection of P and Z^k consists of the point 0 and vertices…

A correspondence between different $Pin$-type structures on a compact surface and quadratic (linear) forms on its homology is constructed. Addition of structures is defined and expressed in terms of these quadratic forms.

We study the question of finding smooth hyperplane sections to a pencil of hypersurfaces over finite fields.

We study genus 1 Gromov-Witten invariants of Fano complete intersections in the projective spaces. Among other things, we show a reconstruction theorem for genus 1 invariants with only ambient insertions, and compute the genus 1 invariants…

In this note, we investigate fibre space structures of a projective irreducible symplectic manifold. We prove that an 2n-dimensional projective irreducible symplectic manifold admits only an n-dimensional fibration over a Fano variety which…