Rational Cuspidal Curves Research Articles

The symplectic isotopy problem for rational cuspidal curves

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 curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.

Symplectic Isotopy of Rational Cuspidal Sextics and Septics

Abstract 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.

Rational cuspidal curves in a moving family of ℙ2

Abstract In this paper we obtain a formula for the number of rational degree d curves in ℙ3 having a cusp, whose image lies in a ℙ2 and that passes through r lines and s points (where r + 2s = 3 d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ2, which has been studied earlier by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]). We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger ([16]) and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali ([1]). We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author ([2]), where they compute the characteristic number of δ-nodal planar curves in ℙ3 with one cusp (for δ ≤ 2).

Kummer surfaces associated with group schemes

We introduce Kummer surfaces $$X={\text {Km}}(C\times C)$$ with the group scheme $$G=\mu _2$$ acting on the self-product of the rational cuspidal curve in characteristic two. The resulting quotients are normal surfaces having a configuration of sixteen rational double points of type $$A_1$$ , together with a rational double point of type $$D_4$$ . We show that our Kummer surfaces are precisely the supersingular K3 surfaces with Artin invariant $$\sigma \le 3$$ , and characterize them by the existence of a certain configuration of thirty curves. After contracting suitable curves, they also appear as normal K3-like coverings for simply-connected Enriques surfaces.

Deformations of plane curves and Jacobian syzygies

AbstractWe relate the equianalytic and the equisingular deformations of a reduced complex plane curve to the Jacobian syzygies of its defining equation. Several examples and conjectures involving rational cuspidal curves are discussed.

Freeness and invariants of rational plane curves

Given a parameterization $\phi$ of a rational plane curve C, we study some invariants of C via $\phi$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $\phi$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $\phi$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational curve can be computed directly from one of its parameterization, without relying on the computation of an equation of C.

Involutive Heegaard Floer homology and rational cuspidal curves

We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.

Free and Nearly Free Curves vs. Rational Cuspidal Plane Curves

We define a class of plane curves that are close to the free divisors in terms of the local cohomology of their Jacobian algebras and such that, conjecturally, any rational cuspidal curve C is either free or belongs to this class. We prove this conjecture when the degree of C is either even or a prime power, or when the group of C is abelian.

Rational cuspidal curves on del-Pezzo surfaces

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 class computation on the moduli space of curves. A topological method is employed in computing the contribution of the degenerate locus to this Euler class.

On the Freeness of Rational Cuspidal Plane Curves

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 particular, we show that this conjecture holds for the curves of degree at most 34.

Rational cuspidal curves in projective surfaces. Topological versus algebraic obstructions

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, Melle–Hernandez and Némethi and is based on the Bézout theorem. The other one is a generalization of the result obtained by Livingston and the author and relies on Ozsváth–Szabó inequalities for [Formula: see text]-invariants in Heegaard Floer homology. We show by means of explicit calculations that the two approaches give very similar obstructions.

The Coolidge–Nagata conjecture

Let E⊆P2 be a complex rational cuspidal curve contained in the projective plane. The Coolidge–Nagata conjecture asserts that E is Cremona-equivalent to a line, that is, it is mapped onto a line by some birational transformation of P2. The second author recently analyzed the log minimal model program run for the pair (X,12D), where (X,D)→(P2,E) is a minimal resolution of singularities, and as a corollary he proved the conjecture in the case when more than one irreducible curve in P2∖E is contracted by the process of minimalization. We prove the conjecture in the remaining cases.

Classification of planar rational cuspidal curves I. C∗∗-fibrations

To classify complex rational cuspidal curves $E\subseteq \mathbb{P}^2$ it remains to classify the ones with complement of log general type, i.e. the ones for which $\kappa(K_X+D)=2$, where $(X,D)$ is a log resolution of $(\mathbb{P}^2,E)$. It is conjectured that $\kappa(K_X+\frac{1}{2}D)=-\infty$ and hence $\mathbb{P}^2\setminus E$ is $\mathbb{C}^{**}$-fibered, where $\mathbb{C}^{**}=\mathbb{C}^1\setminus\{0,1\}$, or $-(K_X+\frac{1}{2}D)$ is ample on some minimal model of $(X,\frac{1}{2}D)$. Here we classify, up to a projective equivalence, those rational cuspidal curves for which the complement is $\mathbb{C}^{**}$-fibered. From the rich list of known examples only very few are not of this type. We also discover a new infinite family of bicuspidal curves with unusual properties.

Heegaard Floer Homologies and Rational Cuspidal Curves. Lecture notes.

This is an expanded version of the lecture course the second author gave at Winterbraids VI in Lille in February 2016. Version 2: revision incorporating referee remarks.

Topological obstructions for rational cuspidal curves in Hirzebruch surfaces

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 theory and is a generalization of a result by Livingston and the first author. The second criterion is obtained by comparing the spectrum of a suitably defined link at infinity of a curve with spectra of its singular points.

Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture

Abstract Let E ⊆ ℙ 2 {E\subseteq\mathbb{P}^{2}} be a complex rational cuspidal curve and let ( X , D ) → ( ℙ 2 , E ) {(X,D)\to(\mathbb{P}^{2},E)} be the minimal log resolution of singularities. We prove that E has at most six cusps and we establish an effective version of the Zaidenberg finiteness conjecture (1994) concerning Eisenbud–Neumann diagrams of E. This is done by analyzing the Minimal Model Program run for the pair ( X , 1 2 ⁢ D ) {(X,\frac{1}{2}D)} . Namely, we show that ℙ 2 ∖ E {\mathbb{P}^{2}\setminus E} is ℂ * * {\mathbb{C}^{**}} -fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

Lattice cohomology and rational cuspidal curves

We show a counterexample to a conjecture of de Bobadilla, Luengo, Melle-Hernandez and Nemethi on rational cuspidal projective plane curves. The counterexample is a tricuspidal curve of degree 8. On the other hand, we show that if the number of cusps is at most 2, then the original conjecture can be deduced from the recent results of Borodzik and Livingston and (lattice cohomology) computations of Nemethi and Roman. Then we formulate a `simplified' (slightly weaker) version, more in the spirit of the motivation of the original conjecture (comparing index type numerical invariants), and we prove it for all currently known rational cuspidal curves. We make all these identities and inequalities more transparent in the language of lattice cohomology of surgery 3-manifolds $S^3_{-d}(K)$, where $K$ is a connected sum of algebraic knots $\{K_i\}_i$. Finally, we prove that the zeroth lattice cohomology of this surgery manifold depends only on the multiset of multiplicities occurring in the multiplicity sequences describing the algebraic knots $K_i$. This result is closely related to the lattice-cohomological reformulation of the above mentioned theorems and conjectures, and provides new computational and comparison procedures.

Free divisors and rational cuspidal plane curves

A characterization of freeness for plane curves in terms of the Hilbert function of the associated Milnor algebra is given as well as many new examples of rational cuspidal curves which are free. Some stronger properties are stated as conjectures.

The Coolidge–Nagata conjecture, part I

Let E⊆P2 be a complex rational cuspidal curve contained in the projective plane and let (X,D)→(P2,E) be the minimal log resolution of singularities. Applying the log Minimal Model Program to (X,12D) we prove that if E has more than two singular points or if D, which is a tree of rational curves, has more than six maximal twigs or if P2∖E is not of log general type then E is Cremona equivalent to a line, i.e. the Coolidge–Nagata conjecture for E holds. We show also that if E is not Cremona equivalent to a line then the morphism onto the minimal model contracts at most one irreducible curve not contained in D.

A Note on the Rational Cuspidal Curves

In this short note we give an elementary combinatorial argument, showing that the conjecture of J. Fernández de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández and A. Némethi [Proc. London Math. Soc. 92 (2006), 99–138, Conjecture 1]