The exceptional locus in the Bertini irreducibility theorem for a morphism

Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field

Let P^n denote the n-dimensional projective space defined over the algebraic closure of a finite field F_q, let V contained P^n be a complete intersection defined over F_q of dimension r and singular locus of dimension at most s, and let π:V-->P^{s+1} be a "generic" linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning π, namely an explicit upper bound of the degree of a proper Zariski closed subset of P^{s+1} which contains all the points defining singular fibers of π. For this purpose we make essential use of the concept of polar variety associated to th…

Irreducibility and stable rationality of the loci of Weierstrass points on curves of genus at most six

We show that for any numerical semigroup H of genus g at most 6, the locus of Weierstrass points on curves of genus g with Weierstrass semigroup H is irreducible and that for all but possibly two semigroups it is stably rational.

Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

We analyse the Gorenstein locus of the Hilbert scheme of $d$ points on $\mathbb{P}^n$ i.e. the open subscheme parameterising zero-dimensional Gorenstein subschemes of $\mathbb{P}^n$ of degree $d$. We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when $d\leq 13$ and find its components when $d = 14$. The proof is relatively self-contained and it does not rely on a computer algebra system. As a by--product, we give equations of the fourth secant variety to the $d$-th Veronese reembedding of $…

Irreducibility of the space of cyclic covers of algebraic curves of fixed numerical type and the irreducible components of $Sing (\bar{\mathfrak M_g})$

We prove irreducibility for the space of cyclic covers of fixed numerical type between smooth projective curves, and also for the space of cyclic covers of prime order and of fixed numerical-combinatorial type between moduli-stable projective curves. As an application we describe the irreducible components of the singular locus of the compactified moduli space of curves $\bar{\mahfrak M_g}$, extending the work of Cornalba, who described the irreducible components of the singular locus of the moduli space of curves $\mahfrak M_g}$.

关于$F$-有理且$F$-纯奇点的Bertini定理

Bertini's theorem for $F$-rational $F$-pure singularities

本文证明特征$p>0$下$F$-有理且$F$-纯拟射影簇与一般超平面的交仍保持这些奇点性质，并给出孤立非$F$-正则的$\mathbb{Q}$-Gorenstein簇特征与指数的关系。

On the irreducibility of secant cones, and an application to linear normality

Let $Y \subset ¶^r$ be a normal nondegenerate m-dimensional subvariety and let $σ(Y)$ denote the maximum dimension of a subvariety $Z \subset Y_{smooth}$ such that $Z$ contains a generic point of some divisor on $Y$ and the tangent planes $T_y Y$ for all $y \in Z$ are contained in a fixed hyperplane. In this article we study the double locus $D \subset $Y$ of its generic projection to $¶^{r-1}$, proving that if the secant variety of $Y$ is the whole space and $σ(Y) < 2m - r + 1$, then $D$ is irreducible. Applying Zak's Tangency theorem we deduce the irreducibility of $D$ when $m > 2(r-1)/3$. T…

闭轨道的密切簇的二次包络的不可约性

Irreducibility of quadratic hulls of osculating varieties of closed orbits

该论文研究连通复半单代数群的闭最高权轨道的p-密切簇的二次包络不可约性，证明当d>4p时二次包络与密切簇重合，确定其不可约性及集合论刻画。

On the locus of smooth plane curves with a fixed automorphism group

In this paper, we study some aspects of the irreducibility of $\widetilde{M_g^{Pl}(G)}$ and its interrelation with the existence of "normal forms", i.e. non-singular plane equations (depending on a set of parameters) such that a specialization of the parameters gives a certain non-singular plane model associated to the elements of $\widetilde{M_g^{Pl}(G)}$. In particular, we introduce the concept of being equation strongly irreducible (ES-Irreducible) for which the locus $\widetilde{M_g^{Pl}(G)}$ is represented by a single "normal form". Henn, and Komiya-Kuribayashi, observed that $\widetilde{…

Explicit Estimates for the Number of Rational Points of Singular Complete Intersections over a Finite Field

Let $V\subset\mathbb{P}^n(\overline{F}_{\hskip-0.7mm q})$ be a complete intersection defined over a finite field $F_{\hskip-0.7mm q}$ of dimension $r$ and singular locus of dimension at most $0\le s\le r-2$. We obtain an explicit version of the Hooley--Katz estimate $||V(F_{\hskip-0.7mm q})|-p_r|=\mathcal{O}(q^{(r+s+1)/2})$, where $|V(F_{\hskip-0.7mm q})|$ denotes the number of $F_{\hskip-0.7mm q}$-rational points of $V$ and $p_r:=|\mathbb{P}^r(F_{\hskip-0.7mm q})|$. Our estimate improves all the previous estimates in several important cases. Our approach relies on tools of classical algebraic…

Geometry of free loci and factorization of noncommutative polynomials

The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.