Cuspidal curves and Heegaard Floer homology

Abstract

Abstract. We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity, the genus, and the degree of the curve; we improve on some degree-multiplicity asymptotic inequalities; finally, we prove some finiteness results, we construct infinite families of examples, and in some cases we give an almost complete classification.

Heegaard Floer Homology and Rational Cuspidal Curves

“…We pass now to the second part of the proof. An analogous argument appeared in [1,4]; we recall it for completeness. The complex CFK´p p Bq is described in Section 4. x a 1 ,...,an b a 1 b¨¨¨b an , where ´1 " 1 P B and 1 " xy P B with gr w " 1 and´1 respectively.…”

“…In particular, we study which Spin c structures on Y extend over CP 2 zN . These computations are slight generalizations of calculations of [1,4,5].…”

A note on cobordisms of algebraic knots

“…Moreover, we also obtain that b + 2 (Z) = 0. The 3-manifold Y n has standard HF ∞ [2,4], and its bottom-most correction terms have been computed in [2,Proposition 4.4] and [4, Theorem 6.10]:…”

Topological obstructions for rational cuspidal curves in Hirzebruch surfaces

“…In order to use this result we need to decrypt the information encoded in inequality (3.2). We will do this in the following steps, following the pattern used in [1,2,3].…”

“…Therefore the equivariant signature is p 1 λ (−1) − p 1 λ (+1). • The Hodge numbers p 1 λ (−1) correspond to values in the spectrum in the interval (0, 1), whereas the Hodge numbers p 1 λ (+1) correspond to values in (1,2). Now let us present a computation of the spectrum.…”