Related papers: Remark on Tono's theorem about cuspidal curves

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…

Fix integers $r\geq 4$ and $i\geq 2$ (for $r=4$ assume $i\geq 3$). Assuming that the rational number $s$ defined by the equation $\binom{i+1}{2}s+(i+1)=\binom{r+i}{i}$ is an integer, we prove an upper bound for the genus of a reduced and…

We derive recursive equations for the characteristic numbers of rational nodal plane curves with at most one cusp, subject to point conditions, tangent conditions and flag conditions, developing techniques akin to quantum cohomology on a…

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.

Given two general rational curves of the same degree in two projective spaces, one can ask whether there exists a third rational curve of the same degree that projects to both of them. We show that, under suitable assumptions on the degree…

We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equirepartition of critical points of a real Lefschetz pencil…

In the first part of this paper, we determine the asymptotic subgroup growth of the fundamental group of a torus knot complement. In the second part, we use this to study random finite degree covers of torus knot complements. We determine…

Using Boij-S\"{o}derberg theory, we give a quick new approach to the results of Han-Kwak and Ahn-Han-Kwak on upper bounds for graded betti numbers of projective schemes in the first nontrivial strand.

Let $C=Z(f)$ be a reduced plane curve of degree $6k$, with only nodes and ordinary cusps as singularities. Let $I$ be the ideal of the points where $C$ has a cusp. Let $\oplus S(-b_i)\to \oplus S(-a_i) \to S\to S/I$ be a minimal resolution…

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RP^n (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those…

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…

Generalizing a classical lemma of Castelnuovo, we characterize rational normal curves (resp. linearly normal elliptic curves) as curves $C\subset \PP^n$ such that the number of linearly independent hypersurfaces $Z\supset C$ of given…

If $c, \overline c\colon [a,b]\to \mathbb R^2$ are two convex planar curve parameterized by affine arc length and $A\colon [a,b]\to [0,\infty)$ is the area bounded by the restriction $c\big|_{[a,s]}$ and the segment between $c(a)$ and…

Let $\mathcal{F}$ be a plane singular curve defined over a finite field $\mathbb{F}_q$. The linear system of plane curves of a given degree passing through the singularities of $\cF$ provides potentially good bounds for the number of points…

Let $X$ be an irreducible smooth geometrically integral projective surface over a field. In this paper we give an effective bound in terms of the Neron--Severi rank $\rho(X)$ of $X$ for the number of irreducible curves $C$ on $X$ with…

We find various lower and upper bounds for the index of Wente tori that contain a continuous family of planar principal curves. We then prove a result that gives an algorithm for computing the index sharply.

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…

For a smooth plane cubic $B$, we count curves $C$ of degree $d$ such that the normalizations of $C\backslash B$ are isomorphic to $\Bbb A^1$, for $d\leq7$ (for $d=7$ under some assumption). We also count plane rational quartic curves…

We prove that the number of directions contained in a set of the form $A \times B \subset AG(2,p)$, where $p$ is prime, is at least $|A||B| - \min\{|A|,|B|\} + 2$. Here $A$ and $B$ are subsets of $GF(p)$ each with at least two elements and…

Using an explicit family of plane quartic curves, we prove the existence of a genus 3 curve over any finite field of characteristic 3 whose number of rational points stays within a fixed distance from the Hasse-Weil-Serre upper bound. We…