The Coolidge–Nagata conjecture, part I

Self Cite

Let E⊆double-struckP2 be a complex curve homeomorphic to the projective line. The Negativity Conjecture asserts that the Kodaira–Iitaka dimension of KX+12D, where false(X,Dfalse)⟶false(P2,Efalse) is a minimal log resolution, is negative. We prove structure theorems for curves satisfying this conjecture and we finish their classification up to a projective equivalence by describing the ones whose complements admit no C∗∗‐fibration. As a consequence, we show that they satisfy the Strong Rigidity Conjecture of Flenner–Zaidenberg. The proofs are based on the almost minimal model program. The obtained list contains one new series of bicuspidal curves.

“…As it was shown in , in this case

κ (K_{X} + \frac{1}{2} D)

plays a crucial role and one can study

\overset{E}{true}

using the modification of the logarithmic Minimal Model Program, the so‐called almost Minimal Model Program (see Section ), applying it to the pair

(X, \frac{1}{2} D)

. The guiding principle is the following conjecture, which strengthens the Coolidge–Nagata conjecture proved recently by Koras and the first author [, ]. Conjecture If

(X, D)

is a log smooth completion of a smooth

double-struckQ

‐acyclic surface then

κ (K_{X} + \frac{1}{2} D) = - \infty

.…”

Section: Resultsmentioning

confidence: 79%

“…We have the following result on chains contractible to smooth points. A similar description was given in [, Lemma 3.7; , Proposition 10]. Lemma For every chain which has a unique

(- 1)

‐curve and is contractible to a smooth point, there is a unique choice of an ordering and unique integers

l ⩾ 0

m_{1}, m_{2}, \dots, m_{l}, x ⩾ 0

, such that the type of the ordered chain is:

\begin{matrix} true \\ left false[{false(2 false)}_{m_{l}}, m_{l - 1} + 3, \dots, m_{2} + 3, {false(2 false)}_{m_{1} + 1}, 1, m_{1} + 3, {false(2 false)}_{m_{2}}, \dots, m_{l} + 3, {false(2 false)}_{x} false], where 2 ∤ l \\ left or false[{false(2 false)}_{m_{l}}, m_{l - 1} + 3, \dots, m_{1} + 3, 1, {false(2 false)}_{m_{1} + 1}, m_{2} + 3, {false(2 false)}_{m_{3}}, \dots, m_{l} + 3, {false(2 false)}_{x} false], where 2 ∣ l . \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 96%

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Classification of planar rational cuspidal curves II. Log del Pezzo models

Palka

Pełka

2019

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“…In fact not many general properties have been proved so far, mostly because the existing theory of log surfaces does not give efficient methods in case

κ = 2

. However, recently M. Koras and the first author proved the Coolidge–Nagata conjecture [, ], which asserts that all rational cuspidal curves are Cremona equivalent to a line. The proof uses, among others, a modification of the log minimal model program (MMP) with half‐integral coefficients, as developed in , and which is based on the generalization of the notion of almost minimality (cf.…”

Section: Resultsmentioning

confidence: 99%

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

Palka

Pełka

2017

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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.

“…On the one hand new algebraic methods have been developed by Koras and Palka. 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 .…”

Section: Introductionmentioning

confidence: 99%

Involutive Heegaard Floer homology and rational cuspidal curves

Borodzik

Hom

Schinzel

2018

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...