Related papers: Factorial threefold hypersurfaces

It is classically known that a real cubic surface in the real projective 3-space cannot have more than one solitary point (locally given by x^2+y^2+z^2=0) whereas it can have up to four nodes (x^2+y^2-z^2=0). We show that on any surface of…

The aim of this paper is sketch a theory of divisibility and factorisation in topological monoids, where finite products are replaced by convergent products. The algebraic case can then be viewed as the special case of discretely…

Let $P$ be a set of $n$ points in real projective $d$-space, not all contained in a hyperplane, such that any $d$ points span a hyperplane. An ordinary hyperplane of $P$ is a hyperplane containing exactly $d$ points of $P$. We show that if…

For any polynomial $P \in \mathbb{C}[X_1,X_2,...,X_n]$, we describe a $\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of $F(P)$…

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with…

Let f : X -> S be any elliptic fibration. If X has dimension 3 and is not uniruled, then X has a minimal model (with terminal singularities) [Mori]. In earlier work we have shown that there exists a birationally equivalent elliptic…

We establish an effective Bertini-type theorem for hypersurfaces $X_f \colon f = 0$ defined over a finite field $k$ for which $f$ has no linear factors over the algebraic closure $\overline{k}$. Given a line $L$ defined over $k$ and a…

We compute the fundamental groups of five maximizing sextics with double singular points only; in four cases, the groups are as expected. The approach used would apply to other sextics as well, given their equations.

Let $Y$ be an algebraic manifold of dimension 3 with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$. Let $X$ be a smooth completion of $Y$ such that the boundary $X-Y$ is the support of an effective…

Let $X$ be a smooth hypersurface $X$ of degree $d\geq4$ in a projective space $\mathbb P^{n+1}$. We consider a projection of $X$ from $p\in\mathbb P^{n+1}$ to a plane $H\cong\mathbb P^n$. This projection induces an extension of function…

We survey a number of results on the counting of points on hypersurfaces defined over finite fields. We also investigate when one can be guaranteed a non-singular point on a projective hypersurface and give a condition on the cardinality of…

We show that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic 2 is at most 12, if the minimal resolution of $X$ is not a…

Let $X$ be an irreducible holomorphic symplectic fourfold and $D$ a smooth hypersurface in $X$. It follows from a result by Amerik and Campana that the characteristic foliation (that is the foliation given by the kernel of the restriction…

In this paper we compute upper bounds for the number of ordinary triple points on a hypersurface in $P^3$ and give a complete classification for degree six (degree four or less is trivial, and five is elementary). But the real purpose is to…

In view of the well-known correspondence between gravitational fields and space-time distributions on a world manifold X, the criterion of gravitation singularities as singularities of these distributions is suggested. In the germ terms,…

In this paper we study holomorphic foliations on $\mathbb{P}^2$ with only one singular point. If the singularity has algebraic multiplicity one, we prove that the foliation has no invariant algebraic curve. We also present several examples…

We provide a formula for the number of $\mathbb{F}_{p}$-points on the Dwork hypersurface $$x_1^n + x_2^n \dots + x_n^n - n \lambda \, x_1 x_2 \dots x_n=0$$ in terms of a $p$-adic hypergeometric function previously defined by the author.…

Let $C$ be a curve in $\mathbb{P}^4$ and $X$ be a hypersurface containing it. We show how it is possible to construct a matrix factorization on $X$ from the pair $(C,X)$ and, conversely, how a matrix factorization on $X$ leads to curves…

We introduce an inductive argument for proving birational superrigidity and K-stability of singular Fano complete intersections of index one, using the same types of information from lower dimensions. In particular, we prove that a…

In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double…