Related papers: Computing a Saito basis from a standard basis

We give a new formula for the rotation number (or Whitney index) of a smooth closed plane curve. This formula is obtained from the winding numbers associated with the regions and the crossing points of the curve. One difference with the…

We study curve singularities in a smooth surface relative to a smooth boundary curve. We consider the semiuniversal deformations and equisingular deformations of curves with a fixed local intersection number $w$ with the boundary, and prove…

These notes are intended as an easy-to-read supplement to part of the background material presented in my talks on enumerative geometry. In particular, the numbers $n_3$ and $n_4$ of plane rational cubics through eight points and of plane…

We consider smooth surfaces $S \subset \Pq$ containing a plane curve $P$ and prove some general result concerning the linear system $|H-P|$. We then look at regular surfaces lying on hypersurfaces of degree $s$ having a plane of…

We study tori attached to the fundamental groups of plane curves with arbitrary singularities. These tori provide complete information about homology of finite abelian covers of the plane branched along the curve. We calculate these tori in…

The discriminant of a smooth plane cubic curve over the complex numbers can be written as a product of theta functions. This provides an important connection between algebraic and analytic objects. In this paper, we perform a new approach…

We use the invariant theory of binary quartics to give a new formula for the Cassels-Tate pairing on the $2$-Selmer group of an elliptic curve. Unlike earlier methods, our formula does not require us to solve any conics. An important role…

Inside the symmetric product of a very general curve, we consider the codimension-one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension. We compute…

We study the Betti tables of reducible algebraic curves with a focus on connected line arrangements and provide a general formula for computing the quadratic strand of the Betti table for line arrangements that satisfy certain hypotheses.…

Logarithmic differential forms and logarithmic vector fields associated to a hypersurface with an isolated singularity are considered in the context of computational complex analysis. As applications, based on the concept of torsion…

A method is proposed for constructing a spline curve of the Bezier type, which is continuous along with its first derivative by a piecewise polynomial function. Conditions for its existence and uniqueness are given. The constructed curve…

We exhibit an algorithm to compute equations of an algebraic curve over a computable characteristic 0 field from the power series expansions of its regular 1-forms at a nonrational point of the curve, extending a 2005 algorithm of Baker,…

Let k be a regular F_p-algebra, let A = k[x,y]/(x^b - y^a) be the coordinate ring of a planar cuspical curve, and let I = (x,y) be the ideal that defines the cusp point. We give a formula for the relative K-groups K_q(A,I) in terms of the…

Let f be a generic polynomial mapping mapping from the plane to the plane. There are constructed quadratic forms whose signatures determine the number of positive and negative cusps of f.

In this paper we consider plane quartics with to involutions. We compute the Dixmier invariants, the bitangents and the Matrix representation problem of these curves, showing that they have symbolic solutions for the last two questions.

A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain…

We target multivariable series associated with resolutions of complex analytic normal surface singularities. In general, the equivariant multivariable analytical and topological Poincar\'e series are well-defined and have good properties…

Let p be a prime integer, and q a power of p. The Ballico-Hefez curve is a non-reflexive nodal rational plane curve of degree q+1 in characteristic p. We investigate its automorphism group and defining equation. We also prove that the…

We consider the identity component of the Sato-Tate group of the Jacobian of curves of the form $$C_1\colon y^2=x^{2g+2}+c, C_2\colon y^2=x^{2g+1}+cx, C_3\colon y^2=x^{2g+1} +c,$$ where $g$ is the genus of the curve and $c\in\mathbb Q^*$ is…

We consider genus 1 enumerative invariants arising from the Smyth-Viscardi moduli space of stable maps from curves with nodes and cusps. We prove that these invariants are equal to the Vakil-Zinger reduced invariants for the quintic…