Related papers: Recovering plane curves from their bitangents

We introduce a notion of good cohomology for multiple lines in $\mathbb{P}^3$ and we classify multiple lines with good cohomology up to multiplicity 4. In particular, we show that the family of space curves of degree d, not lying on a…

The conic sections, as well as the solids obtained by revolving these curves, and many of their surprising properties, were already studied by Greek mathematicians since at least the fourth century B.C. Some of these properties come to the…

It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and…

We bound the dimensions of the graded pieces of the tautological ring of the universal curve from below for genus up to 27 and from above for genus up to 9. As a consequence we obtain the precise structure of the tautological ring of the…

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational…

For each even, positive integer $n$, we define a rational self-map on the space of plane curves of degree $n$, using classical contravariants. In the case of plane quartics, we show that the degree of this map is 15. This answers a question…

Let $\V_{d,n}$ be the Severi variety of irreducible plane curves of degree $d\ge 4$ having $n$ nodes, with $0\le n \le \binom{d-1}{2}-1$. We prove that for every $[\ol C]\in \V_{d,n}$, the infinitesimal variation of the Hodge structure of…

We consider plane curves isomorphic to C*. We prove that with one exception the branches at infinity can be separated by an automorphism of C^2. We also give a bound for selfintersection number of the resolution curve.

In this paper we obtain an explicit formula for the number of curves in a compact complex surface $X$ (passing through the right number of generic points), that has up to one node and one singularity of codimension $k$, provided the total…

We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated to a given non-hyperelliptic curve in terms of geometry of the curve. In order to apply the method, we work out non-hyperelliptic…

In this paper we prove a general theorem concerning the number of translation classes of curves of genus $g$ belonging to a fixed cohomology class in a polarized abelian variety of dimension $g$. For $g = 2$ we recover results of G\"ottsche…

Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. For smooth curves and polygonal lines in the plane, a formula relating the number of…

We are interested in the study of caustics by reflection of irreducible algebraic planar curves (in the complex projective plane). We prove the birationality of the caustic map (for a generic light position). We also give simple formulas…

Explicit models of families of genus 2 curves with multiplication by $\sqrt D$ are known for $D= 2, 3, 5$. We obtain generic models for genus 2 curves over $\mathbb Q$ with real multiplication in 12 new cases, including all fundamental…

The moduli spaces of trigonal curves of odd genus $g>4$ are proven to be rational.

We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of…

We show that the resolution graph of a plane curve singularity admits a canonical decomposition into elementary graphs.

The present paper gives an explicit classification of the isomorphism classes of non-hyperelliptic genus 4 curves over an algebraically closed field of characteristic 0. A non-hyperelliptic genus 4 curve lies on a quadric in $\mathbb{P^3}$…

In this paper, we deal with plane curves with cusps. It is well known that there are various types of cusps. Among them, we investigate criteria for $(n, n+1)$ cusps with respect to several differential conditions and relations between…

We show that the number of tropical curves of given genus and degree through some given general points in the plane does not depend on the position of the points. In the case when the degree of the curves contains only primitive integral…