Related papers: Filling curves for $\mathbb{P}^1 \times \mathbb{P}…

We prove that the moduli space of contact stable maps to $\mathbb{P}^{2n+1}$ of degree $d$ admits a stratification parameterized by graphs. We use it to determine the number of irreducible rational contact curves in $\mathbb{P}^{2n+1}$ with…

Let $p>3$ and consider a prime power $q=p^h$. We completely characterize permutation polynomials of $\mathbb{F}_{q^2}$ of the type $f_{a,b}(X) = X(1 + aX^{q(q-1)} + bX^{2(q-1)}) \in \mathbb{F}_{q^2}[X]$. In particular, using connections…

In order to calculate the multiplicity of an isolated rational curve C in a local complete intersection variety X, i.e. the length of the Hilbert scheme of X at [C], it is important to study infinitesimal neighborhoods of the curve in X.…

Let $P$ and $Q$ be simple polygons with $n$ vertices each. We wish to compute triangulations of $P$ and $Q$ that are combinatorially equivalent, if they exist. We consider two versions of the problem: if a triangulation of $P$ is given, we…

Let $p$ be a large prime, $\ell\geq 2$ be a positive integer, $m\geq 2$ be an integer relatively prime to $\ell$ and $P(x)\in\mathbb{F}_p[x]$ be a polynomial which is not a complete $\ell'$-th power for any $\ell'$ for which…

We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so-called special Picard curves over $\mathbb{Q}$ with good reduction outside 2 and 3, and use this to determine…

Let k be a field of characteristic not 2, let q be a quadratic space over k and let f be an irreducible polynomial with coefficients in k. In 1969, Milnor raised the following question : how can we decide whether q has an isometry with…

We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F_{q^2} whose number of F_{q^2}-rational points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a…

For a non-degenerate irreducible curve $C$ of degree $d$ in $\mathbb{P}^3$ over $\mathbb{F}_q$, we prove that the number $N_q(C)$ of $\mathbb{F}_q$-rational points of $C$ satisfies the inequality $N_q(C) \leq (d-2)q+1$. Our result improves…

Let $X$ be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on $X$, with height at most $B$, for the case in which $X$ is a curve or a surface. In the latter case one excludes…

We give a new proof of the Semistable Reduction Theorem for curves. The main idea is to present a curve $Y$ over a local field $K$ as a finite cover of the projective line $X=\PP^1_K$. By successive blowups (and after replacing $K$ by a…

Let $\mathcal{L}$ be a pencil of plane curves defined over $\mathbb{F}_q$ with no $\mathbb{F}_q$-points in its base locus. We investigate the number of curves in $\mathcal{L}$ whose $\mathbb{F}_q$-points form a blocking set. When the degree…

In Parts I-III we showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ chessboard is a quasipolynomial function of $n$ whose coefficients are essentially polynomials in $q$. In this part…

In this paper, we compute a formula for the $a$-number of curve $\mathbb{X}$ given by the equation $y^{q^2 + q + 1} = x^{q^2 + 1} - x^q$ over the finite field $\mathbb{F}_{q^2}$. The $a$-number is an invariant of the isomorphism class of…

We are concerned with the question of determining the set $C(\mathbb{Q})$, where $C$ is a curve defined by an equation of the form $y^q=f(x)$, where $q$ is an odd prime and $f$ is a polynomial defined over $\mathbb{Q}$. This question can…

We give necessary and sufficient conditions for moduli spaces of semistable chains on a curve to be irreducible and non-empty. This gives information on the irreducible components of the nilpotent cone of GL_n-Higgs bundles and the…

We introduce the notion of a relative of the Hermitian curve of degree $\sqrt{q}+1$ over $\mathbb{F}_q$, which is a plane curve defined by \[(x^{\sqrt{q}}, y^{\sqrt{q}}, z^{\sqrt{q}})A {}^t \!(x,y,z) =0\] with $A \in GL(3, \mathbb{F}_q)$,…

Belyi's theorem asserts that a smooth projective curve $X$ defined over a number field can be realized as a cover of the projective line unramified outside three points. In this short paper we investigate the bejaviour of the minimal degree…

In this paper we give a different proof of Kuz'min's result on the number of irreducible polynomials with the first two coefficients fixed. Our technique is to relate the question to the number of points on a curve, and to calculate the…

We calculate the minimal degree for a class of finite complex reflection groups $G(p,p,q)$, for $p$ and $q$ primes and establish relationships between minimal degrees when these groups are taken in a direct product.