On Rational Cuspidal Projective Plane Curves

Abstract

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

Classification of rational unicuspidal curves with two Newton pairs

“…Moreover, the only known possible local types with two Newton pairs on a rational unicuspidal curve with κ = 2 are those occuring on Orevkov's curves from [14] ((vii) and (viii) in Theorem 3.1.1 below). For those, (2.2.2) was also checked in [7,Theorem 2 (c)]. In particular, all the local types from Theorem 3.1.1 are already explicitly listed in [7] with the appropriate references to their original constructions.…”

“…For those, (2.2.2) was also checked in [7,Theorem 2 (c)]. In particular, all the local types from Theorem 3.1.1 are already explicitly listed in [7] with the appropriate references to their original constructions. …”

“…Therefore, in both cases, κ = −∞. In [7], based on [9] (cf. also [13]), a list of numerical invariants was presented for these curves.…”

“…In [7], a conjecture was formulated by J. Fernández de Bobadilla, I. Luengo, A. Melle-Hernández and A. Némethi on the local topological types of rational cuspidal curves. Moreover, the conjecture was checked explicitly for all cuspidal curves whose complement has logarithmic Kodaira dimension κ ≤ 1 ([7, Theorem 1]) and also for several curves with κ = 2 ([7, Theorem 2]), in particular, for all known rational unicuspidal curves; see also the end of §2.2 and Remark 2.2.6 for further details.…”

Rational cuspidal curves in projective surfaces. Topological versus algebraic obstructions

“…Here (f · C) z i is the local intersection index of the zero locus of f and C at z i . The space H k is a vector subspace of H. We have the following result, proved first in [8]. The present formulation is taken from [6,Lemma 3.17].…”