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

Palka, Karol; Pełka, Tomasz

doi:10.1112/plms.12049

Cited by 16 publications

(68 citation statements)

References 36 publications

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“…In particular, its complement

{double-struckP}^{2} ∖ \bar{E}

double-struckQ

‐acyclic. If this complement is not of log general type then there is a classification, for a summary see, for example, [, Section 2.2] or [, Lemma 2.14] and the references there. In particular, in this case

\overset{E}{true}

has at most two cusps and

{double-struckP}^{2} ∖ \bar{E}

has a

C^{1}

‐ or a

C^{*}

‐fibration [, Proposition 2.6].…”

Section: Resultsmentioning

confidence: 99%

“…For further motivation and evidence toward the Negativity Conjecture, see [, Conjecture 2.5]. In we have classified, up to a projective equivalence, rational cuspidal curves with complements admitting a

C^{* *}

‐fibration, where

{double-struckC}^{* *} = {double-struckC}^{1} ∖ false{0, 1 false}

, in which case Conjecture holds automatically (see [, Lemma 2.4(iii)]). The goal of the current article is to complete the classification, up to a projective equivalence, of rational cuspidal curves for which Conjecture holds.…”

Section: Resultsmentioning

confidence: 99%

“…Theorem Let

\bar{E} \subseteq {double-struckP}^{2}

be a complex curve homeomorphic to

P^{1}

, such that

{double-struckP}^{2} ∖ \bar{E}

is of log general type. Then

{double-struckP}^{2} ∖ \bar{E}

satisfies Negativity Conjecture if and only if either (a)

{double-struckP}^{2} ∖ \bar{E}

has a

C^{* *}

‐fibration, hence

\overset{E}{true}

is of one of the types listed in [, Theorem 1.3], or (b)

\overset{E}{true}

is of one of the types

Q_{3}

Q_{4}

FE (γ)

{scriptFZ}_{2} false(γ false)

H (γ)

scriptI

J (k)

listed in Definition . Each of the above types is realized by a curve which is unique up to a projective equivalence. For a summary of numerical characteristics of the above curves, see Table and [, Table 1]. Note that cases (a) and (b) correspond to the possible outcome of the birational part of the log MMP for

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

, which is a log Mori fiber space over a curve in case (a) and a log del Pezzo surface of Picard rank 1 in case (b), see [, Theorem 4.5(4)].…”

Section: Resultsmentioning

confidence: 99%

“…We summarize these properties in Theorems and , see Section for proofs. We use the names of series

{FZ}_{1}

and

A - G

from and the ones from Definition above. For

k ⩾ 1

we denote by

{scriptOR}_{1} false(k false)

and

{scriptOR}_{2} false(k false)

the curves

C_{4 k}

and

C_{4 k}^{*}

, a part of the series constructed by Orevkov for which the complements are of log general type.…”

Section: Resultsmentioning

confidence: 99%

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

Palka

Pełka

2019

Proc. London Math. Soc.

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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.

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“…In particular, its complement

{double-struckP}^{2} ∖ \bar{E}

double-struckQ

\overset{E}{true}

has at most two cusps and

{double-struckP}^{2} ∖ \bar{E}

has a

C^{1}

‐ or a

C^{*}

‐fibration [, Proposition 2.6].…”

Section: Resultsmentioning

confidence: 99%

C^{* *}

‐fibration, where

{double-struckC}^{* *} = {double-struckC}^{1} ∖ false{0, 1 false}

Section: Resultsmentioning

confidence: 99%

“…Theorem Let

\bar{E} \subseteq {double-struckP}^{2}

be a complex curve homeomorphic to

P^{1}

, such that

{double-struckP}^{2} ∖ \bar{E}

is of log general type. Then

{double-struckP}^{2} ∖ \bar{E}

satisfies Negativity Conjecture if and only if either (a)

{double-struckP}^{2} ∖ \bar{E}

has a

C^{* *}

‐fibration, hence

\overset{E}{true}

is of one of the types listed in [, Theorem 1.3], or (b)

\overset{E}{true}

is of one of the types

Q_{3}

Q_{4}

FE (γ)

{scriptFZ}_{2} false(γ false)

H (γ)

scriptI

J (k)

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

, which is a log Mori fiber space over a curve in case (a) and a log del Pezzo surface of Picard rank 1 in case (b), see [, Theorem 4.5(4)].…”

Section: Resultsmentioning

confidence: 99%

“…We summarize these properties in Theorems and , see Section for proofs. We use the names of series

{FZ}_{1}

and

A - G

from and the ones from Definition above. For

k ⩾ 1

we denote by

{scriptOR}_{1} false(k false)

and

{scriptOR}_{2} false(k false)

the curves

C_{4 k}

and

C_{4 k}^{*}

, a part of the series constructed by Orevkov for which the complements are of log general type.…”

Section: Resultsmentioning

confidence: 99%

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

Palka

Pełka

2019

Proc. London Math. Soc.

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“…Fenske has shown in that the curves in the series

F E (k)

satisfy the weak rigidity conjecture. A stronger conjecture, called the Negativity Conjecture, was proposed by Palka , and has extremely interesting consequences for the study of rational cuspidal plane curves.…”

Section: On the Rational Plane Curves With At Least 3 Cuspsmentioning

confidence: 99%