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We show that a general plane curve of degree at least 4 is uniquely determined by the full set of its bitangent lines. This problem has an elementary solution for degree at least 5, and the paper is almost entirely devoted to curves of…
We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and…
We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.
We give a criterion when a planar tree-like curve, i.e. a generic immersed plane curve each double point of which cuts it into two disjoint parts, can be send by a diffeomorphism of the plane onto a curve with no inflection points. We also…
Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational 2-torsion point on their Jacobian.…
We develop in this article an algorithm that, given a projective curve $C$, computes a \textit{gonal map}, that is, a finite morphism from $C$ to the projective line of minimal degree. Our method is based on the computation of scrollar…
We give necessary and sufficient criteria for a smooth Enriques surface S in P^r to be scheme-theoretically an intersection of quadrics. Moreover we prove in many cases that, when S contains plane cubic curves, the intersection of the…
In mathematics curves are typically defined as the images of continuous real functions (parametrizations) defined on a closed interval. They can also be defined as connected one-dimensional compact subsets of points. For simple curves of…
A point on a plane curve is said to be Galois (for the curve) if the projection from the point as a map from the curve to a line induces a Galois extension of function fields. It is known that the number of Galois points is finite except…
The `linear orbit' of a plane curve of degree $d$ is its orbit in $\P^{d(d+3)/2}$ under the natural action of $\PGL(3)$. In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane…
Let $S$ be a smooth complex minimal surface of general type with $p_g:=h^0(K_S)\ge 4$ whose canonical map is generically finite of odd degree $d>1$ onto a surface $\Sigma$. We assume that the general canonical curve of $S$ is smooth and…
A Brody curve, a.k.a. normal curve, is a holomorphic map from the complex line to the complex projective space of dimension n, such that the family of its translations is normal. We prove that Brody curves omitting n hyperplanes in general…
The canonical degree $C.K_X$ of an integral curve on a smooth projective surface $X$ is conjecturally bounded from above by an expression of the form $A(g-1)+B$, where $g$ is the geometric genus of $C$ and $A$, $B$ are constants depending…
We consider a $C^{1}$ smooth surface with prescribed $p$(or $H$)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed $p$-mean curvature $H\in C^{0},$ we show that any characteristic curve is $C^{2}$ smooth and…
In this paper we study higher Gaussian (or Wahl) maps for the canonical bundle of certain smooth projective curves. More precisely, we determine the rank of higher Gaussian maps of the canonical bundle for plane curves, for curves contained…
This paper considers the task of connecting points on a piece of paper by drawing a curve between each pair of them. Under mild assumptions, we prove that many pairwise disjoint curves are unavoidable if either of the following rules is…
We present four new examples of plane rational curves with two Galois points in positive characteristic, and determine the number of Galois points for three of them. Our results are related to a problem on projective linear groups.
We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves…
We determine the maximum number of rational points on a curve over $\mathbb{F}_2$ with fixed gonality and small genus.
An edge-biregular map arises as a smooth normal quotient of a unique index-two subgroup of a full triangle group acting with two edge-orbits. We give a classification of all finite edge-biregular maps on surfaces of negative prime Euler…