Heegaard Floer Homology and Rational Cuspidal Curves

Borodzik, Maciej; Livingston, Charles

doi:10.1017/fms.2014.28

Cited by 63 publications

(166 citation statements)

References 46 publications

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“…The case g = 0 is excluded in Theorem 1.2: singularities of 1-unicuspidal rational curves have been classified in [5], and the result does not hold in this case. However, applying Theorem 1.1 (which, as pointed out above, is the main theorem of [3]) we can recover the four infinite families of singularities obtained in [5] (see Remarks 6.11 and 6.18). The proof of Theorem 1.2 relies almost exclusively on Theorem 1.1, except when g = 1.…”

Section: Introductionsupporting

confidence: 65%

“…By some small modifications of the argument, we are able to recover (up to finitely many candidates in a bounded region, which after working out a concrete bound, can be checked one by one by computer) the classification result of [5, Theorem 1.1] for g = 0 as well. This is particularly interesting because our method uses the semigroup distribution property of Remark 5.4 only (which, for g = 0 is a result of [3]). After finishing this manuscript, we learned that Tiankai Liu in his PhD thesis [7, Theorem 2.3] among other results also reproved this classification based on the semigroup distribution property only.…”

Section: Unicuspidal Curves With One Puiseux Pairmentioning

confidence: 99%

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Cuspidal curves and Heegaard Floer homology

Bodnár¹,

Celoria²,

Golla³

2016

Proc. London Math. Soc.

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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.

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Section: Introductionsupporting

confidence: 65%

Section: Unicuspidal Curves With One Puiseux Pairmentioning

confidence: 99%

Cuspidal curves and Heegaard Floer homology

Bodnár¹,

Celoria²,

Golla³

2016

Proc. London Math. Soc.

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“…The main result of asserts that the Orevkov curves

C_{4}

and

C_{4}^{*}

, which realize these HN‐types, are the only unicuspidal planar curves with complement of log general type and with one HN‐pair. Using the bounds obtained by Borodzik and Livingstone in via Heegard‐Floer homology methods, Liu [, Theorem 1.1] extended this result by describing possible types of singularities of unicuspidal curves under the assumption that they have at most two Puiseux pairs (see Section 2.4 for definitions). Those which are realized by curves with complements of log general type correspond exactly to the HN‐types

{OR}_{1}

–

{OR}_{2}

.…”

Section: Realization Of Hn‐typesmentioning

confidence: 99%

“…Namely, the minimal set of generators

{\overset{β}{true}}_{0}, \dots, {\overset{β}{true}}_{n}

of this semi‐group is given by (see [, 4.3.5, (4.5), (4.4)]):

{\overset{β}{true}}_{0} = β_{0}, {\overset{β}{true}}_{l + 1} = \frac{1}{e_{l}} false(β_{0} β_{1} + \sum_{i = 1}^{l} e_{i} (β_{i + 1} - β_{i}) false) for l = 0, \dots n - 1,

where

β_{l}

e_{l}

are given by . Let us recall that the Alexander polynomial of the link of

q \in \bar{E}

is given by

Δ false(t false) = 1 + false(t - 1 false) \cdot \sum_{k \notin G (q)} t^{k}

(see [, 2.3]).…”

Section: Appendix Comparison Of Other Numerical Characteristics Of Cmentioning

confidence: 99%

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

Palka

Pełka

2017

Proc. London Math. Soc.

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To classify planar complex rational cuspidal curves E⊆double-struckP2 it remains to classify the ones with complement of log general type, that is, the ones for which κ(KX+D)=2, where (X,D) is a log resolution of (double-struckP2,E). It is conjectured that κ(KX+12D)=−∞ and hence double-struckP2∖E is C∗∗‐fibered, where double-struckC∗∗=double-struckC1∖{0,1}, or −(KX+12D) is ample on some minimal model of (X,12D). Here we classify, up to a projective equivalence, those rational cuspidal curves for which the complement is 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.

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“…In [1], M. Borodzik and Ch. Livingston proved the conjecture for rational unicuspidal curves in the general case ([1, Theorem 1.1]), thus obtaining a necessary combinatorial condition on the numerical invariants of local plane curve singularities occuring on rational unicuspidal curves.…”

mentioning

confidence: 99%