Related papers: On the 10-web by conics on the quartic del Pezzo s…

For $d$ ranging from 2 to 6, we prove that the web by conics naturally defined on any smooth del Pezzo surface of degree $d$ carries an interesting functional identity whose components all are a certain antisymmetric hyperlogarithm of…

In a previous paper, we studied the web by conics $\boldsymbol{\mathcal W}_{{\rm dP}_4}$ on a del Pezzo quartic surface ${\rm dP}_4$ and proved that it enjoys suitable versions of most of the remarkable properties satisfied by Bol's web…

For any $d\in \{1,\ldots,6\}$, we prove that the web of conics on a del Pezzo surface of degree $d$ carries a functional identity whose components are antisymmetric hyperlogarithms of weight $7-d$. Our approach is uniform with respect to…

We prove that a d-web near a point in n-space, where n is greater than 2 and d is greater than 2n-1, is equivalent to an algebraic web, if it has maximal rank or, more generally, if it has (2d - 3n + 1) abelian relations the 1-jets of which…

We discuss the curvilinear web $\boldsymbol{\mathcal W}_{0,n+3}$ on the moduli space $\mathcal M_{0,n+3}$ of projective configurations of $n+3$ points on $\mathbf P^1$ defined by the $n+3$ forgetful maps $\mathcal M_{0,n+3}\rightarrow…

We propose the Legendrian web in a contact three manifold as a second order generalization of the planar web. An Abelian relation for a Legendrian web is analogously defined as an additive equation among the first integrals of its…

Codimension one webs are configurations of finitely many codimension one foliations in general position. Much of the classical theory evolved around the concept of abelian relation: a functional relation among the first integrals of the…

We generalize to webs of any codimension results already known in codimension one. Given a holomorphic $d$-web $\cal W$ of codimension $q$ $(q\leq n-1)$ in an ambiant $n$-dimensional holomorphic manifold $U$, we define for any integer $p$…

We explain a classical construction of a del Pezzo surface of degree d = 4 or 5 as a smooth order two congruence of lines in 3-space whose focal surface is a quartic surface $X_{20-d}$ with 20-d ordinary double points. We also show that…

Some classes of cubic fourfolds are birational to fibrations over $P^2$, where the fibers are rational surfaces. This is the case for cubics containing a plane (resp. an elliptic ruled surface), where the fibers are quadric surfaces (resp.…

Let $d\geq3$ be an integer. For a holomorphic $d$-web $\mathcal{W}$ on a complex surface $M$, smooth along an irreducible component $D$ of its discriminant $\Delta(\mathcal{W}),$ we establish an effective criterion for the holomorphy of the…

For any fixed $1 \leq \ell \leq 9$, we characterize all Wahl singularities that appear in degenerations of del Pezzo surfaces of degree $\ell$. This extends the work of Manetti and Hacking-Prokhorov in degree $9$, where Wahl singularities…

Three-dimensional del Pezzo varieties of degree 2 are double covers of projective space $\mathbb{P}^{3}$ branced in a quadric. In this paper we prove that if a del Pezzo variety of degree 2 has exactly 15 nodes then the corresponding…

In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that…

Let $S$ be an elliptic surface over a smooth curve $C$ with a section $O$. We denote its generic fiber by $E_S$. For a divisor $D$ on $S$, we canonically associate a $C(C)$-rational point $P_D$. In this note, we give a description of $P_D$…

We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type A the dimension of non-conical del Pezzo varieties is bounded by $12…

Let $(X,D)$ be an open log del Pezzo surface of rank one, that is, $X$ is a normal projective surface of Picard rank one, the boundary $D$ is a reduced nonzero divisor on $X$, and the anti-log canonical divisor $-(K_X+D)$ is ample. We show…

We exhibit new examples of rational cubic fourfolds, parametrized by a countably infinite union of codimension-two subvarieties in the moduli space. Our examples are fibered in sextic del Pezzo surfaces over the projective plane; they are…

We classify webs of minimal degree rational curves on surfaces and give a criterion for webs being hexagonal. In addition, we classify Neron-Severi lattices of real weak del Pezzo surfaces. These two classifications are related to root…

We present a new normal form for cubic surfaces that is well suited for p-adic geometry, as it reveals the intrinsic del Pezzo combinatorics of the 27 trees in the tropicalization. The new normal form is a polynomial with eight terms,…