Smoothable del Pezzo surfaces with quotient singularities

Hacking, Paul; Prokhorov, Yuri

doi:10.1112/s0010437x09004370

Cited by 79 publications

(104 citation statements)

References 16 publications

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“…Our main result characterises mutations between triangles; thus we characterise certain deformations, over P 1 , with fibers given by fake weighted projective planes. We recover and generalise certain results of Hacking and Prokhorov [HP10,Theorem 4.1] connecting the fake weighted projective planes with T -singularities to solutions of Markov-type equations. We prove the following: Proposition 1.1.…”

Section: Introductionsupporting

confidence: 77%

Mutations of Fake Weighted Projective Planes

Coates

Gonshaw

Kasprzyk

et al. 2015

Proceedings of the Edinburgh Mathematical Society

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Abstract. In previous work by Coates, Galkin, and the authors, the notion of mutation between lattice polytopes was introduced. Such a mutation gives rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterisation of such mutations in terms of T -singularities. We show also that the weights involved satisfy Diophantine equations, generalising results of Hacking-Prokhorov.

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Section: Introductionsupporting

confidence: 77%

Mutations of Fake Weighted Projective Planes

Coates

Gonshaw

Kasprzyk

et al. 2015

Proceedings of the Edinburgh Mathematical Society

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“…In Sect. 6.3, we point out that the Markov type I equations have appeared before, in relation to 3-blocks exceptional collections in the del Pezzo surfaces [20] and Q-Gorenstein smoothing of weighted projective spaces to del Pezzo surfaces [16]. We ask if there is a correspondence between ATBDs, 3-blocks exceptional collections and Q-Gorenstein degenerations of a given del Pezzo surface (see Questions 6.2, 6.3).…”

Section: To Mutually Different Hamiltonian Isotopy Classesmentioning

confidence: 94%

“…In [16,Theorem 1.2], it is shown that the weighted projective planes that admit a Q-Gorenstein smoothing are precisely the ones given by the limit orbifolds of the ATBDs obtained by total mutation of the ones in Fig. 1.…”

Section: Question 62 Suppose We Have An Atbd With Node Typementioning

confidence: 99%

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Vianna

2017

Sel. Math. New Ser.

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We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for CP 2 #CP 2 , CP 2 #2CP 2 , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in CP 2 #kCP 2 for k = 0, 3, 4, 5, 6, 7, 8. We name these tori. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that CP 2 #CP 2 also has infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for CP 2 #2CP 2 . Finally, the Lagrangian tori n 1 ,n 2 ,n 3 p,q,r

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“…Here n(l) is the total number of LDP-polygons, m(l) is the number of LDPtriangles (i.e. rank one toric log del Pezzo surfaces), n T (l) is the number of LDP-polygons with T-singularities [20], and m T (l) is the number of LDPtriangles with T-singularities. For fixed index l, it is possible to classify all LDP-polygons [27].…”

Section: Fano Polygonsmentioning

confidence: 99%