Related papers: Rational cuspidal curves with four cusps on Hirzeb…
In this article we give an upper bound for the number of cusps on a cuspidal curve on a Hirzebruch surface. We adapt the results that have been found for a similar question asked for cuspidal curves on the projective plane, and restate the…
We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer…
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…
Submission on request. This Master thesis from 2008 (University of Oslo, Norway) contains no new results, but it provides an overview of plane rational cuspidal curves, in particular curves of low degree. Please note that new results on…
In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by…
In this paper, we consider rational cuspidal plane curves having exactly two cusps whose complements have logarithmic Kodaira dimension two. We classify such curves with the property that the strict transforms of them via the minimal…
In this paper we obtain a formula for the number of rational degree d curves in $\mathbb{P}^3$ having a cusp, whose image lies in a $\mathbb{P}^2$ and that passes through $r$ lines and $s$ points (where $r + 2s = 3d + 1$). This problem can…
We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed…
In this paper, we consider rational cuspidal plane curves having at least three cusps. We give an upper bound of the self-intersection number of the proper transforms of such curves via the minimal embedded resolution of the cusps. For a…
We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler…
We classify rational cuspidal curves of degrees 6 and 7 in the complex projective plane, up to symplectic isotopy. The proof uses topological tools, pseudoholomorphic techniques, and birational transformations.
In the previous paper [E-print alg-geom/9507004] we classified the rational cuspidal plane curves C with a cusp of multiplicity deg C - 2. In particular, we showed that any such curve can be transformed into a line by Cremona…
We give several constructions of bicuspidal rational complex projective plane curves, and list the Newton pairs and the multiplicity sequences of the singularities on the resulting curves. Although the existence of some of the listed cusp…
We solve the problem of computing characteristic numbers of rational space curves with a cusp, where there may or may not be a condition on the node. The solution is given in the form of effective recursions. We give explicit formulas when…
A new, simple method to approach enumerative questions about rational curves on rational surfaces is described. Applications include a short proof of Kontsevich's formula for plane curves and a the solution of the analogous problem for the…
Some classes of cubic fourfolds are birational to fibrations over $P^2$, where the fibers are rational surfaces. This is the case for cubics containing a plane (resp. an elliptic ruled surface), where the fibers are quadric surfaces (resp.…
We survey some results on real rational surfaces focused on their topology and their birational geometry.
We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo,…
We study the geometry and codes of quartic surfaces with many cusps. We apply Gr\"obner bases to find examples of various configurations of cusps on quartics.
We bring additional support to the conjecture saying that a rational cuspidal plane curve is either free or nearly free. This conjecture was confirmed for curves of even degree, and in this note we prove it for many odd degrees. In…