The geometry of sporadic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math>-embeddings into <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>

Koras, Mariusz; Palka, Karol; Russell, Peter B.

doi:10.1016/j.jalgebra.2016.03.001

Cited by 7 publications

(17 citation statements)

References 12 publications

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“…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%

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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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“…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%

“…Such images are classified in [5,16] in case when they are smooth and in [4] under some regularity conditions.…”

Section: It Follows Thatmentioning

confidence: 99%

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

Palka

Pełka

2019

Proc. London Math. Soc.

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“…Remark It is possible, and in some situations more convenient to allow more general choices of

Z_{1}

Y_{j}

for

j ⩾ 1

, which do not satisfy (running inductive arguments more smoothly is one of the reasons). For instance, instead of we could ask

Y_{j}

to satisfy

p_{j} ⩽ c_{j}

(as it is done in ), which results in replacing all pairs

((), \binom{c_{j}}{p_{j}})

where

p_{j} > c_{j}

with sequences

{((), \binom{c_{j}}{c_{j}})}_{r_{j}} (\frac{0pt}{c_{j}})

, where

r_{j}

is the integer part of

p_{j} / c_{j}

and

{(\frac{0pt}{c})}_{r}

means

((), \binom{c}{c})

repeated

r

times. Additionally, we could allow a more general choice of

Z_{1}

, which will relax the condition

p_{1} ∤ c_{1}

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Note that after the first three blowups, the cusp meets the last exceptional curve with multiplicity 2. Therefore (see the discussion preceding ), if we used HN‐pairs as in instead of the standard ones, namely, if we required that

p_{2} ⩽ c_{2}

and thus chose some general smooth germ

Y_{2}

transversal to the last exceptional curve instead of the maximally tangent one, then the HN‐type would be

(\frac{0pt}{6}) (\frac{0pt}{2}) (\frac{0pt}{2})

. The fact that we have two Puiseux pairs (equivalently, that the Puiseux sequence is of length 3) corresponds to the fact that the exceptional divisor over

q_{1}

has one branching component.…”

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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“…The interesting case of Theorem 1, for k " C, is thus the description of AutpA 2 , Γq when Γ is isomorphic to C˚(or a union of such curves). Note that there exist only partial classifications of the closed embeddings of C˚into A 2 , which are moreover very involved (see [CNKR09], [BŻa10] and more recently [KPR14]). Hence, one cannot check a few cases to derive Theorem 1.…”

Section: Introductionmentioning

confidence: 99%