Related papers: Ciliberto-Di Gennaro conjecture for sextic hypersu…

Gersho's conjecture in 3D asserts the asymptotic periodicity and structure of the optimal centroidal Voronoi tessellation. This relatively simple crystallization problem remains to date open. We prove bounds on the geometric complexity of…

Let Q -> B be a quadric fibration and T -> B a family of sextic du Val del Pezzo surfaces. Making use of the recent theory of noncommutative mixed motives, we establish a precise relation between the Schur-finiteness conjecture for Q, resp.…

A very general hypersurface of dimension $n$ and degree $d$ in complex projective space is rational if $d \leq 2$, but is expected to be irrational for all $n, d \geq 3$. Hypersurfaces in weighted projective space with degree small relative…

We investigate Manin's conjecture for del Pezzo surfaces of degree five with a conic bundle structure, proving matching upper and lower bounds, and the full conjecture in the Galois general case.

We study equisingular deformation problems for curves and surfaces in algebraic families, with particular emphasis on situations where nodal behavior is no longer generic. Extending classical Severi theory, we develop deformation--theoretic…

For a projective variety $Z$ and for any integer $p$, define the $p$-th N\'eron-Severi group $NS_p(Z)$ of $Z$ as the image of the cycle map $A_{p}(Z)\to H_{2p}(Z; \mathbb{C})$. Now let $X\subset \Ps^{2m+1}$ ($m\geq 1$) be a projective…

Recently, Marian-Oprea-Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Johnson, they provided a conjectural correspondence between…

It is shown that hypersurfaces of degree $M$ in ${\mathbb P}^M$, $M\geqslant 5$, with at most quadratic singularities of rank at least 3, satisfying certain conditions of general position, are birationally superrigid Fano varieties and the…

Catanese and Tonoli showed that the maximal cardinality for an even set of nodes on a sextic surface is 56 and they constructed such nodal surfaces. In this paper we give an alternative, rather simple, construction for these surfaces…

We prove the statement in the title, solving in this way a conjecture stated by Ginot for manifolds with corners. Along the way, we establish a derived Swiss-cheese additivity theorem and an alternative proof for the hyperdescent of…

Given a complex curve C of genus 2, there is a well-known relationship between the moduli space of rank 3 semistable bundles on C and a cubic hypersurface known as the Coble cubic. Some of the aspects of this is known to be related to the…

Several papers have been written studying unexpected hypersurfaces. We say a finite set of points Z admits unexpected hypersurfaces if a general union of fat linear subspaces imposes less that the expected number of conditions on the ideal…

We construct algebraic surfaces with a large number of type A singularities. Bivariate polynomials presented in previous works for the construction of nodal surfaces and certain families of Belyi polynomials are used. In some cases explicit…

This is an expository paper based on the results in [12] and [16]. The main goal is to prove the following two conjectures for genus up to two. (1) Witten's conjecture on the relations between higher spin curves and Gelfand-Dickey…

The codimension-three conjecture states that any regular holonomic module extends uniquely beyond an analytic subset with codimension equal to or larger than three. We give a proof of this conjecture.

We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\Bbb C^n$, then any local diffeomorphism $F:X \to \Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover…

We introduce the notion of biconservative hypersurfaces, that is hypersurfaces with conservative stress-energy tensor with respect to the bienergy. We give the (local) classification of biconservative surfaces in 3-dimensional space forms.

For a codimension 1 holomorphic foliation $\mathcal F$ on $\mathbb P_{\mathbb C}^{n}$ satisfying reasonable assumptions, there are estimations of the degree of invariant hypersurfaces H in terms of the degree of $\mathcal F$ (Carnicer,…

In this note we will review the most important results and questions related to Chern conjecture and isoparametric hypersurfaces, as well as their interactions and applications to various aspects in mathematics.

The more recent paper "Generic strange duality for K3 surfaces" by the authors contains stronger results.