Related papers: The Coolidge-Nagata conjecture, part I

Consider a smooth, geometrically irreducible, projective curve of genus $g \ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that…

In this short note we give an elementary combinatorial argument, showing that the Conjecture of J. Fern\'andez de Bobadilla, I. Luengo, A. Melle-Hern\'andez, A. N\'emethi follows from the results of M. Borodzik and C. Livingston in the case…

We study singular rational curves in projective space, deducing conditions on their parametrizations from the value semigroups $\sss$ of their singularities. In particular, we prove that a natural heuristic for the codimension of the space…

We generalize Miyanishi's theory of almost minimal models of log smooth surfaces with reduced boundary to the case of arbitrary log surfaces defined over an algebraically closed field. Given an MMP run of a log surface $(X,D)$ we define and…

In this paper we discuss some variations of Nagata's conjecture on linear systems of plane curves. The most relevant concerns non-effectivity (hence nefness) of certain rays, which we call \emph{good rays}, in the Mori cone of the blow-up…

We bring additional support to the conjecture saying that a rational cuspidal plane curve is either free or nearly free. This conjecture was confirmed for curves of even degree, and in this note we prove it for many odd degrees. In…

By Mazur's Torsion Theorem, there are fourteen possibilities for the non-trivial torsion subgroup $T$ of a rational elliptic curve. For each $T$, such that $E$ may have additive reduction at a prime $p$, we consider a parameterized family…

Modifying an approach of J. Roe, this paper gives an improved lower bound on the degrees d such that for general points p1,...,pn in P2 and m > 0 there is a plane curve of degree d vanishing at each point pi with multiplicity at least m. In…

We consider the set E of curves with positive algebraic curvature, whose extremities and tangents in their extremities are given. For each of the curves of E, we define the minimum of the radius of curvature. We first prove that there…

In this article we prove a finiteness result on the number of log minimal models for $3$-folds in char $p>5$. We then use this result to prove a version of Batyrev's conjecture on the structure of nef cone of curves on $3$-folds in…

Consider an elliptic curve $E$ over a number field $K$. Suppose that $E$ has supersingular reduction at some prime $\mathfrak{p}$ of $K$ lying above the rational prime $p$. We completely classify the valuations of the $p^n$-torsion points…

We use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a klt pair $(X,\Delta)$ can be detected on the base of the $(K_{X}+\Delta)$-trivial reduction map. Thus we show that…

We study the interaction between the group law on an elliptic curve and the additive structure of $x$-coordinates of rational points on an elliptic curve. Let $E/\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r \geq 1$, $d \geq 1$…

For every smooth Jordan curve $\gamma$ and rectangle $R$ in the Euclidean plane, we show that there exists a rectangle similar to $R$ whose vertices lie on $\gamma$. The proof relies on Shevchishin's theorem that the Klein bottle does not…

Chisini's conjecture asserts that for a cuspidal curve $B\subset \mathbb P^2$ a generic morphism $f$ of a smooth projective surface onto $\mathbb P^2$ of degree $\geq 5$, branched along $B$, is unique up to isomorphism. We prove that if…

In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of…

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get for any…

We give lower bounds for the degree of the discriminant with respect to y of separable polynomials f in K[x,y] over an algebraically closed field of characteristic zero. Depending on the invariants involved in the lower bound, we give a…

A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally…

Let $X$ be a smooth projective split horospherical variety over a number field $k$ and $x\in X(k)$. Contingent on Vojta's conjecture, we construct a curve $C$ through $x$ such that (in a precise sense) rational points on $C$ approximate $x$…