Related papers: Transverse linear subspaces to hypersurfaces over …

We show that the complement of a degree $d$ hypersurface in a projective complete intersection, whose defining equations have degrees strictly larger than $d$, has a rational connectivity higher than expected. The key new feature is that a…

Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional…

In this paper we give a complete local parametric classification of the hypersurfaces with dimension at least three of a space form that carry a totally geodesic foliation of codimension one. A classification under the assumption that the…

Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number…

We show various properties of smooth projective D-affine varieties. In particular, any smooth projective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic zero such D-affine…

We show that if $M$ is an Einstein hypersurface in an irreducible Riemannian symmetric space $\overline{M}$ of rank greater than $1$ (the classification in the rank-one case was previously known), then either $\overline{M}$ is of noncompact…

Asgarli, Ghioca, and Reichstein proved that if $K$ is a field with $|K|>2$, then for any positive integers $d$ and $n$, and separable field extension $L/K$ with degree $m=\binom{n+d}{d}$, there exists a point $P\in \mathbb{P}^n(L)$ which…

This is a continuation of "Rational curves on hypersurfaces of low degree", math.AG/0203088. We prove that if d^2+d+1 < n and d > 2, then for a general hypersurface X_d in P^n of degree d, for each degree e the space of rational curves of…

Alex Waldron proved that for sufficiently general degree $d$ hypersurfaces in projective $n$-space, the Fano scheme parameterizing $r$-dimensional linear spaces contained in the hypersurface is nonempty precisely for the degree range $n\geq…

This article presents the theory of focal locus applied to the hypersurfaces in the projective space which are (finitely) covered by linear spaces and such that the tangent space is constant along these spaces.

We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the…

V. Arnold's problem 1987-14 asks whether there exist smooth hypersurfaces in $R^N$ (other than the conics in odd-dimensional spaces) for which the volume of the segment cut by any hyperplane from the body bounded by such a hypersurface is…

This paper investigates the density of hypersurfaces in a projective normal simplicial toric variety over a finite field having a quasismooth intersection with a given quasismooth subscheme. The result generalizes the formula found by B.…

We study projective surfaces $X \subset \mathbb{P}^r$ (with $r \geq 5$) of maximal sectional regularity and degree $d > r$, hence surfaces for which the Castelnuovo-Mumford regularity $\reg(\mathcal{C})$ of a general hyperplane section…

This article proves hypersurfaces of degree d in projective n-space are "rationally simply-connected" if $d^2 \leq n$. In a forthcoming paper, de Jong and I prove a slightly weaker result when $d^2 \leq n+1$.

We classify hypersurfaces of the Minkowski space $\L^{n+1}$ that carry a totally geodesic foliation with complete leaves of codimension one. We prove that such a hypersurface is ruled, or a partial tube over a curve or contains a two or…

In the toric variety $\mathcal{T}$, with Cox ring graded by $\deg(z_{2i})=(1,-1,0)$, $\deg(z_{2i+1})=(1,0,-1)$ and $\deg(w_\pm)=(0,1,0),(0,0,1)$, we study hypersurfaces $\widetilde{X}^{2n}\subset\mathcal T$ of multidegree $(2d+1,-d,-d)$…

Let D = {D_{1},...,D_{l}} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let \Omega^{1}_{P^n}(log D) be the logarithmic bundle attached to it. Following [1], we show that…

In this note we look at the freeness for complex affine hypersurfaces. If $X \subset \mathbb{C}^n$ is such a hypersurface, and $D$ denotes the associated projective hypersurface, obtained by taking the closure of $X$ in $\mathbb{P}^n$, then…

We show that given a smooth projective variety X over C with dim(X) > 2, an ample line bundle O(1) on X and an integer n > 1, any n distinct points on a generic hypersurface of degree d in X are linearly independent in CH_0(X) if d >> 0.…