Related papers: On subsets of the normal rational curve

We study embedded rational curves in projective toric varieties. Generalizing results of the first author and Zotine for the case of lines, we show that any degree $d$ rational curve in a toric variety $X$ can be constructed from a special…

Let $\mathbb{F}_q$ be a finite field with $q=p^{e}$ elements, where $p$ is a prime number and $e \geq 1$ is an integer. In this paper, by means of generalized Reed-Solomon (GRS) codes, we construct two new classes of quantum…

We show that there is a bound depending only on g and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g-3. If K = Q, an…

A {\em generalized hyperfocused arc} $\mathcal H $ in $PG(2,q)$ is an arc of size $k$ with the property that the $k(k-1)/2$ secants can be blocked by a set of $k-1$ points not belonging to the arc. We show that if $q$ is a prime and…

Let $\mathbb{F}_q$ be a field with $q$ elements. In this note, we study some generalized arcs, that is, sets of $\mathbb{F}_q$-points in the projective plane $\mathbb{P}^2(\mathbb{F}_q)$ such that no six of them are on a conic. First, we…

We study rational surfaces having an even set of disjoint $(-4)$-curves. The properties of the surface $S$ obtained by considering the double cover branched on the even set are studied. It is shown, that contrarily to what happens for even…

We prove that the moduli space C(d) of plane curves of degree d (for projective equivalence) is rational except possibly if d= 6, 7, 8, 11, 12, 14, 15, 16, 18, 20, 23, 24, 26, 32, 48.

We determine the splitting (isomorphism) type of the normal bundle of a generic genus-0 curve with 1 or 2 components in any projective space, as well as the (sometimes nontrivial) way the bundle deforms locally with a general deformation of…

We show that if F is the rational numbers or a multiquadratic number field, p is 2,3, or 5, and K/F is a Galois extension of degree a power of p, then for elliptic curves E/Q ordered by height, the average dimension of the p-Selmer groups…

We study the maximum length of $q$-ary codes as a function of alphabet size, code size, and Singleton defect. For an $(n, M, d)_q$ code with dimension $\kappa = \log_q M \ge 2$ and Singleton defect $s = n - \lceil\kappa\rceil + 1 - d$, we…

Let $A$ be an elliptic curve over $\Q$ of square free conductor $N$. We prove that if $A$ has a rational torsion point of prime order $r$ such that $r$ does not divide $6N$, then $r$ divides the order of the cuspidal subgroup of $J_0(N)$.

We study the rationality of the components of the conchoid to an irreducible algebraic affine plane curve, excluding the trivial cases of the isotropic lines, of the lines through the focus and the circle centered at the focus and radius…

For $r \geq 1$ an odd integer, we provide a sequence from the function field $\mathcal{F}_{q, r}$ of the maximal curve over $\mathbb{F}_{q^{2r}}$ defined by the affine equation $y^q+y=x^{q^r + 1}$. This sequence has high nonlinear…

In the paper \cite{Lau16}, it was shown that the restriction of a pseudoeffective divisor $D$ to a subvariety $Y$ with nef normal bundle is pseudoeffective. Assuming the normal bundle is ample and that $D|_Y$ is not big, we prove that the…

A partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ is a collection of $(k-1)$-dimensional subspaces with trivial intersection, i.e., each point is covered at most once. So far the maximum size of a partial $(k-1)$-spread in…

For each odd prime power q, we construct an infinite sequence of rational functions f(X) in F_q(X), each of which is exceptional, which means that for infinitely many n the map c-->f(c) induces a bijection of P^1(F_{q^n}). Moreover, each of…

We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map,…

We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We…

A hyperelliptic curve over $\mathbb Q$ is called "locally soluble" if it has a point over every completion of $\mathbb Q$. In this paper, we prove that a positive proportion of hyperelliptic curves over $\mathbb Q$ of genus $g\geq 1$ are…

Let $p$ be a rational prime, $q>1$ a power of $p$ and $F=\mathbb{F}_q(t)$. For an integer $d\geq 2$, let $D$ be a central division algebra over $F$ of dimension $d^2$ which is split at $\infty$ and has invariant $\mathrm{inv}_x(D)=1/d$ at…