Complex algebraic plane curves via Poincaré-Hopf formula. II. Annuli

Borodzik, Maciej; Żołądek, Henryk

doi:10.1007/s11856-010-0013-1

Cited by 15 publications

(54 citation statements)

References 16 publications

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“…Type

scriptH

corresponds to [, Theorem 8.2(ii.3)]. See Remarks , (a)–(c) and (a) for a comparison with the conditional classification of Borodzik and Żoła̧dek .…”

Section: Resultsmentioning

confidence: 99%

“…Remark (a)If

A

is as in Lemma (d) then

\bar{E} ∖ π_{0} false(A false) \subseteq {double-struckP}^{2} ∖ π_{0} false(A false)

is the image of a proper injective morphism

{double-struckC}^{*} ⟶ {double-struckC}^{2}

. Such images are classified in in case when they are smooth and in under some regularity conditions. (b)If

A^{'}

is as in Lemma (g) then

π_{0} false(A^{'} false)

is a line which is a good asymptote in the sense of for the above

C^{*}

‐embedding. …”

Section: Possible Types Of Cuspsmentioning

confidence: 99%

“…Remark (a)Let

ι_{3}, ι_{4} : {double-struckC}^{*} ⟶ {double-struckC}^{2}

be the injective morphisms given, respectively, by (v) and (w) in . Via an automorphism of

C^{2}

they are equivalent to

\bar{E} ∖ {scriptl}_{1} ↪ {double-struckP}^{2} ∖ {scriptl}_{1}

, where

\overset{E}{true}

is of type

Q_{3}

Q_{4}

, respectively, and

l_{1}

is the line tangent to

q_{1} \in \bar{E}

.…”

Section: Existence and Uniquenessmentioning

confidence: 99%

“…We leave the details to the reader. Remark (a)A curve

\overset{E}{true}

of type

{scriptFZ}_{2} false(k + 2 false)

for

k ⩾ 2

can be also obtained as the closure of the image of a singular embedding

{double-struckC}^{*} ⟶ {double-struckC}^{2}

given by [, (g)] via the standard embedding

{double-struckC}^{2} ∋ false(x, y false) \mapsto false[x : y : 1 false] \in {double-struckP}^{2}

. Then the line

{z = 0}

at infinity is the line

l_{12}

joining

q_{1}, q_{2} \in \bar{E}

.…”

Section: Existence and Uniquenessmentioning

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.

show abstract

“…Type

scriptH

corresponds to [, Theorem 8.2(ii.3)]. See Remarks , (a)–(c) and (a) for a comparison with the conditional classification of Borodzik and Żoła̧dek .…”

Section: Resultsmentioning

confidence: 99%

“…Remark (a)If

A

is as in Lemma (d) then

\bar{E} ∖ π_{0} false(A false) \subseteq {double-struckP}^{2} ∖ π_{0} false(A false)

is the image of a proper injective morphism

{double-struckC}^{*} ⟶ {double-struckC}^{2}

. Such images are classified in in case when they are smooth and in under some regularity conditions. (b)If

A^{'}

is as in Lemma (g) then

π_{0} false(A^{'} false)

is a line which is a good asymptote in the sense of for the above

C^{*}

‐embedding. …”

Section: Possible Types Of Cuspsmentioning

confidence: 99%

“…Remark (a)Let

ι_{3}, ι_{4} : {double-struckC}^{*} ⟶ {double-struckC}^{2}

be the injective morphisms given, respectively, by (v) and (w) in . Via an automorphism of

C^{2}

they are equivalent to

\bar{E} ∖ {scriptl}_{1} ↪ {double-struckP}^{2} ∖ {scriptl}_{1}

, where

\overset{E}{true}

is of type

Q_{3}

Q_{4}

, respectively, and

l_{1}

is the line tangent to

q_{1} \in \bar{E}

.…”

Section: Existence and Uniquenessmentioning

confidence: 99%

“…We leave the details to the reader. Remark (a)A curve

\overset{E}{true}

of type

{scriptFZ}_{2} false(k + 2 false)

for

k ⩾ 2

can be also obtained as the closure of the image of a singular embedding

{double-struckC}^{*} ⟶ {double-struckC}^{2}

given by [, (g)] via the standard embedding

{double-struckC}^{2} ∋ false(x, y false) \mapsto false[x : y : 1 false] \in {double-struckP}^{2}

. Then the line

{z = 0}

at infinity is the line

l_{12}

joining

q_{1}, q_{2} \in \bar{E}

.…”

Section: Existence and Uniquenessmentioning

confidence: 99%

See 2 more Smart Citations

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

Palka

Pełka

2019

Proc. London Math. Soc.

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show abstract

“…For the description of the polynomial

{\tilde{q}}_{k}

we use the formulas given in [, Main Theorem (h)], following the suggestion kindly offered to us by Karol Palka. Define first a sequence of polynomials