Related papers: Intersections of three quadrics in $\mathbb{P}^7$

A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible…

We classify three-dimensional nodal Fano varieties that are double covers of smooth quadrics branched over intersections with quartics acted on by finite simple non-abelian groups, and study their rationality.

We show that if X is a smooth rationally connected threefold and C is a smooth projective curve then C can be embedded in X. Furthermore, a version of this property characterises rationally connected varieties of dimension at least 3. We…

Complete intersections may be unexpectedly simple over fields of positive characteristic: for instance, they may be unirational despite being of general type. One explanation is given by profiles, structure that tracks the special shape of…

A Kloosterman refinement for function fields $K=\mathbb{F}_q(t)$ is developed and used to establish the quantitative arithmetic of the set of rational points on a smooth complete intersection of two quadrics $X\subset \mathbb{P}^{n-1}_{K}$…

Let $Y$ be a smooth complete intersection of a quadric and a cubic in $\mathbb{P}^n$, with $n$ even. We show that $Y$ has a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of (powers…

We study linear systems cut out by cones of fixed degree on a smooth complex curve $C\subset\mathbb{P}^{3}$. We develop a systematic study of the families of such systems, considering their limits, their infinitesimal behaviour and some…

In this paper we give a complete characterization of the intersections between the Norm-Trace curve over $\mathbb{F}_{q^3}$ and the curves of the form $y=ax^3+bx^2+cx+d$, generalizing a previous result by Bonini and Sala, providing more…

We describe a new method of constructing rational surfaces with given invariants in P^4 and present a family of degree 11 rational surfaces of sectional genus 11 with 2 six-secants that we found with this method.

A general net of quadric surfaces, together with a choice of a base point, defines a net of plane cubics via the Gale transformation of the remaining seven base points. To both nets, one can also naturally associate the same smooth plane…

We complete the study of rationality problem for hypersurfaces $X_t\subset \mathbb{P}^4$ of degree $4$ invariant under the action of the symmetric group $S_6$.

We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a…

A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by…

We construct the first non-trivial examples of complete families of non-degenerate smooth space curves, and show that the base of such a family cannot be a rational curve. Both results rely on the study of the strong semistability of…

We study the connectedness of the real locus of smooth geometrically rational Fano threefolds and prove a sufficient criterion of $\mathbb{R}$-rationality.

In this paper we characterize smooth complex projective varieties that admit a quadric bundle structure on some dense open subset in terms of the geometry of certain families of rational curves.

We develop explicit techniques to investigate algebraic quasi-hyperbolicity of singular surfaces through the constraints imposed by symmetric differentials. We apply these methods to prove that rational curves on Barth's sextic surface,…

We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either…

We study rationality questions for Fano schemes of linear spaces on smooth complete intersections of two quadrics, especially over non-closed fields. Our approach is to study hyperbolic reductions of the pencil of quadrics associated to…

Let C be a smooth cubic curve in the complex projective plane. We show that for every positive integer k, there are only finite number of rational curves of degree k each intersects the cubic C at exactly one point. The number of such…