Number of singular points of an annulus in \mathbb{C}^2

Borodzik, Maciej; Żołądek, Henryk

doi:10.5802/aif.2650

Cited by 3 publications

(28 citation statements)

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“…As 4 is even and 4 − 1 is odd, Theorem 6.4 below will not obstruct the existence of a (3, 6e + 10) on a (4, 4) curve, regardless of the parity of e. (6,6) with one cusp with one single Puiseux pair in X 0 . We now compute the spectra for the link at infinity and the link at the cusp for rational unicuspidal curves of type (6,6) with a single Puiseux pair in X 0 . We do this for all curves with Puiseux pair that fits the genus formula, and there are three such curves.…”

Section: 1mentioning

confidence: 94%

“…Let C be a unicuspidal curve of type (6,6) in X 0 , and let the cusp have a single Puiseux pair. We have a = b = 6, g = (6 − 1)(6 − 1) = 25 and 2g = 50.…”

Section: Curves Of Typementioning

confidence: 99%

“…We have a = b = 6, g = (6 − 1)(6 − 1) = 25 and 2g = 50. The list of theoretically possible Puiseux pairs is (2, 51), (3,26) and (6,11) 6,11 ∩ (x, x + 1) = 34. This curve passes the criterion from Theorem 1.3, and it can in fact be constructed by a simple transformation of the plane unicuspidal curve y 5 z − x 6 = 0.…”

Section: Curves Of Typementioning

confidence: 99%

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Topological obstructions for rational cuspidal curves in Hirzebruch surfaces

Borodzik¹,

Moe²

2016

Michigan Math. J.

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Abstract. We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard-Floer theory and is a generalization of a result by Livingston and the first author. The second criterion is obtained by comparing the spectrum of a suitably defined link at infinity of a curve with spectra of its singular points.

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Section: 1mentioning

confidence: 94%

“…Let C be a unicuspidal curve of type (6,6) in X 0 , and let the cusp have a single Puiseux pair. We have a = b = 6, g = (6 − 1)(6 − 1) = 25 and 2g = 50.…”

Section: Curves Of Typementioning

confidence: 99%

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Topological obstructions for rational cuspidal curves in Hirzebruch surfaces

Borodzik¹,

Moe²

2016

Michigan Math. J.

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“…Example 2.1.7. Let C and C be two rational cuspidal curves of degree 5 with the same cuspidal configuration [2 2 ], [3,2], given by the parametrizations (s 5 :…”

Section: Inflection Points On Plane Cuspidal Curvesmentioning

confidence: 99%

“…The question of the number of cusps on a rational curve has also been studied for other curves than projective rational curves. Borodzik and Zoladek study this question in [3] for plane algebraic annuli, that is, reduced algebraic curves C ⊂ C 2 that are homeomorphic to C * . They believe that their methods can be extended to all rational curves on C 2 , but claim that the computations in that situation are highly complex.…”

Section: Corollary 2429 (Corollary To Theorem 235)mentioning

confidence: 99%

Segre classes on smooth projective toric varieties

Moe

Qviller

2013

Math. Z.

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Produced in cooperation with Akademika publishing. The thesis is produced by Akademika publishing merely in connection with the thesis defence. Kindly direct all inquiries regarding the thesis to the copyright holder or the unit which grants the doctorate. Writing this thesis has been a long lasting and challenging task. At the time of its completion it is time to thank everyone who has contributed. First, I would like to thank Professor Ragni Piene for giving me a solid training in mathematics, for introducing me to the puzzling problem of cuspidal curves, and for providing the ideas, suggestions and advice necessary to complete this thesis. I am very grateful for everything that you have helped me achieve the past ten years and for the support and care you have provided along the way. This thesis could not have been written without the encouragement I have received from my colleagues in the algebra group. In particular, I have to thank Nikolay Qviller for inspirational, instructive and productive cooperation. I am very much indebted to Georg Muntingh for interesting discussions and invaluable contributions, for helping me with images, and for proofreading. Moreover, I would like to thank Heidi Camilla Mork and Professor Kristian Ranestad for motivation and support. I am also very grateful to Inger Christin Borge and Arne Bernhard Sletsjøe for always believing in me and for including me in amazing teaching projects. Additionally, I want to thank Jørgen Vold Rennemo and John Christian Ottem for sharing enthusiasm for mathematics and for impressing me on an everyday basis. I also want to thank Professor Keita Tono for helpful explanations, and Torsten Fenske for sending me a copy of his PhD-thesis. My PhD-position has been funded by the Institute of Mathematics, University of Oslo, and I have been associated to the Centre of Mathematics for Applications (CMA). I thank you for giving me the opportunity to write this thesis. Additionally, I want to thank everyone in the administration and IT-drift. Through the years I have met a vast number of fantastic people in the corridors of the Niels Henrik Abel building. In particular, I would like to thank Agnieszka,

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Remark on Tono's theorem about cuspidal curves

Orevkov

2017

Mathematische Nachrichten

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Number of singular points of an annulus in \mathbb{C}^2

Cited by 3 publications

References 6 publications

Topological obstructions for rational cuspidal curves in Hirzebruch surfaces

Topological obstructions for rational cuspidal curves in Hirzebruch surfaces

Segre classes on smooth projective toric varieties

Remark on Tono's theorem about cuspidal curves

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