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Heegaard Floer homology and plane curves with non-cuspidal singularities

Maciej Borodzik, Beibei Liu, Ian Zemke

TL;DR

The paper develops a Floer-theoretic framework to constrain singular configurations of complex curves in $\mathbb{C}P^2$ by connecting the $H_{1}$-action on knot Floer complexes of knotifications to link Floer actions, and by expressing border data through tubular neighborhoods described via link surgery. Central to the approach is the knotification formalism, the computation of $V_s$-invariants and $d$-invariants via large surgery, and the use of staircase complexes for L-space knots, together with explicit analyses of the Hopf link, $T(2,2n)$ torus links, and the Borromean knot. The authors derive concrete genus–double-point obstructions and $A_n$-singularity bounds for nonrational non-cuspidal curves, including obstructions for trading genus for double points. Their results hinge on precise tensor-product calculations, the notion of split towers, and the Levine–Ruberman $d$-invariant framework, enabling explicit, computable obstructions for families of singularities and offering insights relevant to line arrangements and algebraic-type versus smooth-type models.

Abstract

We study possible configurations of singular points occuring on general algebraic curves in $\mathbb{C}P^2$ via Floer theory. To achieve this, we describe a general formula for the $H_{1}$-action on the knot Floer complex of the knotification of a link in $S^3$, in terms of natural actions on the link Floer complex of the original link. This result may be interest on its own.

Heegaard Floer homology and plane curves with non-cuspidal singularities

TL;DR

The paper develops a Floer-theoretic framework to constrain singular configurations of complex curves in by connecting the -action on knot Floer complexes of knotifications to link Floer actions, and by expressing border data through tubular neighborhoods described via link surgery. Central to the approach is the knotification formalism, the computation of -invariants and -invariants via large surgery, and the use of staircase complexes for L-space knots, together with explicit analyses of the Hopf link, torus links, and the Borromean knot. The authors derive concrete genus–double-point obstructions and -singularity bounds for nonrational non-cuspidal curves, including obstructions for trading genus for double points. Their results hinge on precise tensor-product calculations, the notion of split towers, and the Levine–Ruberman -invariant framework, enabling explicit, computable obstructions for families of singularities and offering insights relevant to line arrangements and algebraic-type versus smooth-type models.

Abstract

We study possible configurations of singular points occuring on general algebraic curves in via Floer theory. To achieve this, we describe a general formula for the -action on the knot Floer complex of the knotification of a link in , in terms of natural actions on the link Floer complex of the original link. This result may be interest on its own.

Paper Structure

This paper contains 36 sections, 42 theorems, 173 equations, 5 figures.

Table of Contents

  1. Introduction
  2. General context
  3. Overview of the paper
  4. Applications and perspectives
  5. Organization
  6. Singular curves in the smooth category
  7. Review of Heegaard Floer theory
  8. Heegaard Floer complexes with multiple basepoints
  9. The link Floer complex
  10. Homological actions
  11. Heegaard Floer homology of a knotification
  12. Generalized correction terms of Levine and Ruberman
  13. $V$-invariants
  14. Large surgery formula
  15. Topology of complex curves and their neighborhoods
  16. ...and 21 more sections

Key Result

Theorem 1

Let $C$ be a reduced curve of degree $d$ and genus $g$. Suppose that $C$ has cuspidal singular points $z_1,\dots,z_\nu$, whose semigroup counting functions are $R_1,\dots,R_\nu$, respectively. Assume that apart from these $\nu$ points, the curve $C$ has, for each $n\ge 1$, $m_n\ge 0$ singular points For any $k=1,\dots,d-2$, we have: Here $R$ denotes the infimal convolution of the functions $R_1,\

Figures (5)

  • Figure 1: An unkot with 4 basepoints. The dashed arc is $\lambda$.
  • Figure 2: The configuration of the band $B$, the basepoints, and the arc $\lambda'\subset \mathbb{L}'$.
  • Figure 3: A genus $0$ Heegaard diagram for the Hopf link. The thick (red) curve is the $\alpha$-curve, the thin (blue) curve is the $\beta$-curve. The dotted curve is used to compute the action of $H_1(S^2\times S^1;\mathbb Z)$ on the knotification of the Hopf link.
  • Figure 4: A Heegaard diagram for $T(2,4)$ from a doubly pointed open book. The dashed line is an arc $\lambda$ connecting $w_1$ and $w_2$.
  • Figure 5: The chain complexes for $T(2,4)$ (1st level from top) and $T(2,6)$ (2nd level). On the third level is the map $A_\lambda$ on the complex for $T(2,6)$, and on the bottom is the map $\Phi_{w_2}$.

Theorems & Definitions (71)

  • Theorem : see Theorem \ref{['thm:genus_and_double']}
  • Theorem : Theorem \ref{['thm:double_neg']}
  • Example 1.1: see Example \ref{['ex:orevkov_neg']}
  • Example 1.2: see Example \ref{['ex:fg_cases']}
  • Definition 1.3
  • Definition 1.4
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.9: Knotification
  • ...and 61 more