Explicit estimates for polynomial systems defining irreducible smooth complete intersections

This paper deals with properties of the algebraic variety defined as the set of zeros of a "deficient" sequence of multivariate polynomials. We consider two types of varieties: ideal-theoretic complete intersections and absolutely irreducible varieties. For these types, we establish improved bounds on the dimension of the set of deficient systems of each type over an arbitrary field. On the other hand, we establish improved upper bounds on the number of systems of each type o...

We determine upper bounds on the number of rational points of an affine or projective algebraic set defined over an extension of a finite field by a system of polynomial equations, including the case where the algebraic set is not defined over the finite field by itself. A special attention is given to irreducible but not absolutely irreducible algebraic sets, which satisfy better bounds. We study the case of complete intersections, for which we give a decomposition, coarser ...

We study the set of common F_q-rational zeros of systems of multivariate symmetric polynomials with coefficients in a finite field F_q. We establish certain properties on these polynomials which imply that the corresponding set of zeros over the algebraic closure of F_q is a complete intersection with "good" behavior at infinity, whose singular locus has a codimension at least two or three. These results are used to estimate the number of F_q-rational points of the correspond...

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 esti...

Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random ...

Let $ e_1, ..., e_c $ be positive integers and let $ Y \subseteq \mathbb{P}^n$ be the monomial complete intersection defined by the vanishing of $x_1^{e_1}, ..., x_c^{e_c}$. In this paper we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points $Hilb^d(Y)$, and discuss applications to the Hilbert scheme of points $Hilb^d(X)$ of arbitrary complete intersections $X \subseteq \mathbb{P}^n$.

In this paper, we consider a problem of counting multiplicities. We fix a counting function of multiplicity of rational points in a hypersurface of a projective space over a finite field, and we give an upper bound for the sum with respect to this counting function in terms of the degree of the hypersurface, the dimension and the cardinality of the finite field. This upper bound gives a description of the complexity of the singular locus of this hypersurface. In order to obta...

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 \pi:V-->P^{s+1} be a "generic" linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning \pi, namely an explicit upper bound of the degree of a proper Zariski closed subset of P^{s+1} which contains all the p...

We prove that the space of smooth rational curves of degree $e$ in a general complete intersection of multidegree $(d_1, ..., d_m)$ in $\PP^n$ is irreducible of the expected dimension if $\sum_{i=1}^m d_i <\frac{2n}{3}$ and $n$ is large enough. This generalizes the results of Harris, Roth and Starr \cite{hrs}, and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if $\sum_{i=1}^m d_i$ is small c...

We show explicit estimates on the number of $q$--rational points of an $F_q$--definable affine absolutely irreducible variety of the algebraic closure of the finite field $F_q$ of $q$ elements. Our estimates for a hypersurface significantly improve previous estimates of W. Schmidt and M.-D. Huang and Y.-C. Wong, while in the case of a variety our estimates improve those of S. Ghorpade and G. Lachaud in several important cases. Our proofs rely on elementary methods of effectiv...