Smoothings and Rational Double Point Adjacencies for Cusp Singularities

Engel, Philip; Friedman, Robert

doi:10.48550/arxiv.1609.08031

Cited by 1 publication

(2 citation statements)

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“…(1), ( 2) and (3) hold if and only if EF16], Proposition 1.5). Let (Y gen , D gen ) be a generic log Calabi-Yau surface, where D gen has at least three boundary components.…”

Section: Toolsmentioning

confidence: 97%

“…Cusp singularities come in dual pairs such that the links are diffeomorphic but have opposite orientations. If (Y , p) is obtained by contracting the boundary of a log Calabi-Yau surface (Y, D) to a cusp singularity p ∈ Y , then, conjecturally, (Y, D) corresponds to an irreducible component of the deformation space of the dual cusp ( [GHK15a], [E15], [EF16]). This is expected as a consequence of mirror symmetry: Y \D is mirror to the Milnor fiber of the corresponding smoothing of the dual cusp ( [Ke15], [HKe21]).…”

Section: Introductionmentioning

confidence: 99%

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A cone conjecture for log Calabi-Yau surfaces

Li¹

2022

Preprint

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We consider log Calabi-Yau surfaces (Y, D) with singular boundary. In each deformation type, there is a distinguished surface (Y e , D e ) such that the mixed Hodge structure on H 2 (Y \ D) is split. We prove that (1) the action of the automorphism group of (Y e , D e ) on its nef effective cone admits a rational polyhedral fundamental domain; and (2) the action of the monodromy group on the nef effective cone of a very general surface in the deformation type admits a rational polyhedral fundamental domain. These statements can be viewed as versions of the Morrison cone conjecture for log Calabi-Yau surfaces. In addition, if the number of components of D is ≤ 6, we show that the nef cone of Y e is rational polyhedral and describe it explicitly. This provides infinite series of new examples of Mori Dream Spaces.

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“…(1), ( 2) and (3) hold if and only if EF16], Proposition 1.5). Let (Y gen , D gen ) be a generic log Calabi-Yau surface, where D gen has at least three boundary components.…”

Section: Toolsmentioning

confidence: 97%