Related papers: Bertini irreducibility theorems over finite fields

A projective hypersurface $X \subseteq \mathbb P^n$ has defect if $h^i(X) \neq h^i(\mathbb P^n)$ for some $i \in \{n, \dots, 2n-2\}$ in a suitable cohomology theory. This occurs for example when $X \subseteq \mathbb P^4$ is not $\mathbb…

Let $X$ be an irreducible projective variety and $f$ a morphism $X \rightarrow \mathbb{P}^n$. We give a new proof of the fact that the preimage of any linear variety of dimension $k\ge n+1-\dim f(X)$ is connected. We prove that the…

A key property of an algebraic variety is whether it is absolutely irreducible, meaning that it remains irreducible over the algebraic closure of its defining field, and determining absolute irreducibility is important in algebraic geometry…

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

We study the degree of irreducible morphisms in any Auslander-Reiten component of a finite dimensional algebra over an algebraically closed field. We give a characterization for an irreducible morphism to have finite left (or right) degree.…

Poonen and Slavov recently developed a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing. In this paper, we extend their work by proving an analogous bound for the dimension of the…

We give an inductive proof that the generalized Severi varieties -- the varieties which parametrize (irreducible) plane curves of given degree and genus, with a fixed tangency profile to a given line at several general fixed points and…

Let $X$ be a minuscule homogeneous space, an odd quadric, or an adjoint homogenous space of type different from $A$ and $G_2$. Le $C$ be an elliptic curve. In this paper, we prove that for $d$ large enough, the scheme of degree $d$…

We explore the existence of irreducible and reducible arc-sections in an irreducible hypersurface singularity germ along finite projections. In particular we provide examples of irreducible isolated hypersurface singularities for which no…

We study irreducible subvarieties of the universal hypersurface $\mathcal{X}/B$ of degree $d$ and dimension $n$. We prove that when $d$ is sufficiently large, a degree $kd$ subvariety $Z$ which dominates $B$ comes from intersection with a…

We prove that, under mild restrictions, the space of codimension-one foliations of degree one on a smooth projective complete intersection has two irreducible components of logarithmic type. We also prove that the same conclusion holds for…

Let $k$ be an algebraically closed field and let $b$ and $n$ be integers with $n\geq 3$ and $1\leq b \leq n-1.$ Consider the moduli space $X$ of hypersurfaces in $\mathbb{P}^n_k$ of fixed degree $l$ whose singular locus is at least…

We study the problem of the irreducibility of the Hessian variety $\mathcal{H}_f$ associated with a smooth cubic hypersurface $V(f)\subset \mathbb{P}^n$. We prove that when $n\leq5$, $\mathcal{H}_f$ is normal and irreducible if and only if…

Let n > 2 and let d < (n+1)/2. We prove that for a general hypersurface X of degree d in P^n, all the genus 0 Kontsevich moduli spaces M_{0,n}(X,e) are irreducible, reduced, local complete intersection stacks of the expected dimension.

Let $k$ be an algebraically closed field of characteristic $p > 0$. We show that if $X\subseteq\mathbb{P}^n_k$ is an equidimensional subscheme with Hilbert--Kunz multiplicity less than $\lambda$ at all points $x\in X$, then for a general…

We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $f(T_1,\ldots, T_n,X)$ is an irreducible polynomial over the field of rational functions over a finite field $\mathbb{F}_q$ of characteristic $p$,…

Let $k$ be an algebraically closed field. Fix integers $n$ and $b$ with $n\geq 3$ and $1\leq b\leq n-1.$ Let $T^d_k$ be the moduli space of hypersurfaces $[F]$ in $\mathbb{P}^n_k$ of degree $l$ whose singular locus contains a subscheme of…

Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique,…

We study the question of finding smooth hyperplane sections to a pencil of hypersurfaces over finite fields.

We use function field analytic number theory to establish the irreducibility and dimension of the moduli space that parameterises morphisms of fixed degree from $\mathbb{P}^2$ to an arbitrary smooth hypersurface of sufficiently small…