Remark on Tono's theorem about cuspidal curves

Orevkov, Stepan

doi:10.1002/mana.201500178

Cited by 2 publications

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“…A positive answer to this question follows from the work of Tono [58, 4.4], cf. [36], who bounded the number of twigs of

D

by 17 (see Subsection 2.1 for definitions). Its stronger version, Conjecture 1.1(c), was proved by Palka [41, 1.3] in case when

X\setminus D

is a complement of a rational cuspidal curve, that is,

X\setminus D\cong \mathbb {P}^2\setminus \overline{E}

for some curve

\overline{E}

that is homeomorphic to

\mathbb {P}^1

(in the Euclidean topology).…”

Section: Introductionmentioning

confidence: 99%

Smooth Q$\mathbb {Q}$‐homology planes satisfying the negativity conjecture

Pełka

2023

Journal of London Math Soc

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A complex algebraic surface is a ‐homology plane if for . The Negativity Conjecture of Palka asserts that , where is a log smooth completion of . We give a complete description of smooth ‐homology planes satisfying the Negativity Conjecture. We restrict our attention to those of log general type, as otherwise their geometry is well‐understood. We show that, as conjectured by tom Dieck and Petrie, they can be arranged in finitely many discrete series, each obtained in a uniform way from an arrangement of lines and conics on . We infer that these surfaces satisfy the Rigidity Conjecture of Flenner and Zaidenberg; and a conjecture of Koras, which asserts that .

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“…A positive answer to this question follows from the work of Tono [58, 4.4], cf. [36], who bounded the number of twigs of

D

by 17 (see Subsection 2.1 for definitions). Its stronger version, Conjecture 1.1(c), was proved by Palka [41, 1.3] in case when

X\setminus D

is a complement of a rational cuspidal curve, that is,

X\setminus D\cong \mathbb {P}^2\setminus \overline{E}

for some curve

\overline{E}

that is homeomorphic to

\mathbb {P}^1

(in the Euclidean topology).…”

Section: Introductionmentioning

confidence: 99%

Smooth Q$\mathbb {Q}$‐homology planes satisfying the negativity conjecture

Pełka

2023

Journal of London Math Soc

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“…These methods, based on the minimal model program, have led to a solution of the Cooligde–Nagata conjecture and to a proof of an effective (much stronger than the original) version of the Zajdenberg finiteness conjecture; see . As it was noted later by Orevkov , the Zaidenberg conjecture in the original version can be inferred from the results of Tono . There is a work in progress on giving a full classification, at least assuming the rigidity conjecture; see for details.…”

Section: Introductionmentioning

confidence: 99%

Involutive Heegaard Floer homology and rational cuspidal curves

Borodzik

Hom

Schinzel

2018

Proc. London Math. Soc.

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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.Theorem 4.10. Let C be a rational cuspidal curve of odd degree with two singular points z 1 and z 2 . Let K 1 and K 2 be links of singularities of z 1 and z 2 . Then at least one of K 1 and K 2 is an even L-space knot.We illustrate the above application by a simple example. It is a well-known result (see [18, Section 6.1.3]) but we give the first topological proof.Example 1.3. A rational cuspidal curve of degree 5 cannot have two singular points with Puiseux sequences (2; 11) and (2; 3). It also cannot have two singular points with Puiseux sequences (2; 7) and (2; 7).A rational cuspidal curve with Puiseux sequences (2; 9) and (2; 5) actually exists. It is item 7 in the list [19, Theorem 2.3.10]. See also [18, Section 6.1.3].This degree 5 example is quite remarkable from the following point of view. In [3] Bodnár and Némethi noted that the criterion of [4] does not actually restrict singular points, but only so-called multiplicity sequences. We do not give all the details, but point out that the criterion of [4] is unable to distinguish the case of singular points (2; 11), (2; 3) and (2; 9), (2; 5). We give more examples in Section 5. 1.3. Outline of the proof The main idea of the proof comes from [4], although technical problems already appear at an early stage. Consider a rational cuspidal curve C ⊂ CP 2 and let N be a tubular neighborhood of C. Let M = ∂N and set W = CP 2 \ N . As in [4] we identify M with a surgery on the connected sum of links of singularities of C. Moreover H k (W ; Q) = 0 for k > 0. The latter fact implies by [22] that for any Spin c structure s on M that extends to W we have d(M, s) = 0, where d is the Ozsváth-Szabó d-invariant. The equality d(M, s) = 0 was exploited in [4].In the present article we rely on a result of Hendricks and Manolescu, that for any Spin structure s on M that extends over W we have d(M, s) = d(M, s) = 0, where d and d are the invariants defined in [14]; see Section 2.1. We look at the canonical Spin structure on M , that is, the one corresponding to m = 0; see Section 2.2 for notation. The problem is that the canonical Spin structure extends over W if and only if deg C is odd. Therefore our results are restricted to curves of odd degree; see Section 4.1.If C has one singular point, then M is an L-space and it follows from [14, Section 4.4] that d(M, s) = d(M, s) = d(M, s). In particular, our result says nothing new for rational cuspidal curves with one singular point. However, if C has more than one singular point, the condition d(M, s) = d(M, s) becomes restrictive. The second, and actually, more difficult, part of the paper translates the equality d(M, s) = d(M, s) into a tractable condition on semigroups of singular points of C.Throughout, we let F = Z/2Z. In...

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Remark on Tono's theorem about cuspidal curves

Abstract: We give an upper bound for the number of cusps of a plane affine or projective curve via its first Betti number. K E Y W O R D SPlane algebraic curve, singular point M S C ( 2 0 1 0 ) 14H50Lin and Zaidenberg [4,5] (see also [1, §5], [3], and Remark 2 below) asked the following questions.

Cited by 2 publications

References 7 publications

Smooth Q$\mathbb {Q}$‐homology planes satisfying the negativity conjecture

Smooth Q$\mathbb {Q}$‐homology planes satisfying the negativity conjecture

Involutive Heegaard Floer homology and rational cuspidal curves

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