Related papers: On Gonality and Canonical Models of Unicuspidal Ra…

Let $C$ be an integral and projective curve; and let $C'$ be its canonical model. We study the relation between the gonality of $C$ and the dimension of a rational normal scroll $S$ where $C'$ can lie on. We are mainly interested in the…

Let $C$ be an integral and projective curve whose canonical model $C'$ lies on a rational normal scroll $S$ of dimension $n$. We mainly study some properties on $C$, such as gonality and the kind of singularities, in the case where $n=2$…

We completely classify all plane curves of degree at most 30 with a unique cuspidal (locally unibranch) singular point and rational normalization in terms of the Newton pairs parameterizing the cusp. We distinguish between prime and…

We give refined statements and modern proofs of Rosenlicht's results about the canonical model C' of an arbitrary complete integral curve C. Notably, we prove that C and C' are birationally equivalent if and only if C is nonhyperelliptic,…

Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$…

We study the gonality of curves $C$ over $\mathbb C$ whose normalization is composed of one or two copies of $\mathbb P^1$. In the first case, $C$ is a nodal curve with $g(C)$ nodes, and in the second case $C$ is a so-called binary curve.…

By developing a suitable version of the circle method, we show that the space of degree $e$ rational curves on a smooth hypersurface of degree $d$ has only canonical singularities provided its dimension is sufficiently large with respect to…

Let $C \subset P^{g-1}$ be a smooth canonical curve of genus $g \geq 3$. The purpose of this article is to further develop a method to classify varieties having $C$ as their curve section, using Gaussian map computations. In a previous…

Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very difficult…

In this thesis we give a complete description of the syzygies of irreducible, nonsingular, canoncial curves $C$ of genus 9. This includes a collection of all possible Betti tables for $C$. Moreover a direct correspondence between these…

We obtain new examples and the complete list of the rational cuspidal plane curves $C$ with at least three cusps, one of which has multiplicity ${\rm deg}\,C - 2$. It occurs that these curves are projectively rigid. We also discuss the…

Let C be a smooth complex projective curve of genus g and let X be its second symmetric product. This paper concerns the study of some attempts at extending to X the notion of gonality. In particular, we prove that the degree of…

The gonality conjecture predicts that the gonality of a curve can be read off Koszul cohomology of line bundles of sufficiently large degree. We verify this conjecture for generic curves of odd genus. The even-genus case was previously…

We extend the computations from our previous paper arXiv:2005.07054 to determine the maximum number of rational points on a curve over $\mathbb{F}_3$ and $\mathbb{F}_4$ with fixed gonality and small genus. We find, for example, that there…

Let C be an ACM (projectively normal) nondegenerate smooth curve in projective 3-space, and suppose C is general in its Hilbert scheme - this is irreducible once the postulation is fixed. Answering a question posed by Peskine, we show the…

We determine the maximum number of rational points on a curve over $\mathbb{F}_2$ with fixed gonality and small genus.

We prove some lower bounds on certain nonegative twists of the canonical bundle of a subvariety of a generic hypersurface in projective space. In particular we prove that the generic sextic threefold contains no rational or elliptic curves…

We consider the parameterization ${\mathbf{f}}=(f_0,f_1,f_2)$ of a plane rational curve $C$ of degree $n$, and we want to study the singularities of $C$ via such parameterization. We do this by using the projection from the rational normal…

It is a very old and interesting open problem to characterize those collections of embedded topological types of local plane curve singularities which may appear as singularities of a projective plane curve C of degree d. The goal of the…

This paper concerns the existence of curves with low gonality on smooth hypersurfaces of sufficiently large degree. It has been recently proved that if $X\subset \mathbb{P}^{n+1}$ is a hypersurface of degree $d\geq n+2$, and if $C\subset X$…