6,404 research outputs found
Mirror symmetry for log Calabi-Yau surfaces I
We give a canonical synthetic construction of the mirror family to a pair
(Y,D) of a smooth projective surface with an anti-canonical cycle of rational
curves, as the spectrum of an explicit algebra defined in terms of counts of
rational curves on Y meeting D in a single point. In the case D is
contractible, the family gives a smoothing of the dual cusp, and thus a proof
of Looijenga's 1981 cusp conjecture.Comment: 144 pages, 3 figures, Second version significantly shorter, 109
pages. The first version has a lot of material (particularly in the
introduction and material on cyclic quotient singularities) which does not
appear in the new version. Download version 1 if this material is desired.
Third and final version, small changes from Version 2, to appear in Publ.
IHE
Moduli of surfaces with an anti-canonical cycle
We prove a global Torelli theorem for pairs (Y,D), where Y is a smooth
projective rational surface and D is an effective anti-canonical divisor which
is a cycle of rational curves. This Torelli theorem was conjectured by Friedman
in 1984. In addition, we construct natural universal families for such pairs.Comment: Final version. Much simplified proofs. To appear in Compositi
Birational geometry of cluster algebras
We give a geometric interpretation of cluster varieties in terms of blowups
of toric varieties. This enables us to provide, among other results, an
elementary geometric proof of the Laurent phenomenon for cluster algebras (of
geometric type), extend Speyer's example of an upper cluster algebra which is
not finitely generated, and show that the Fock-Goncharov dual basis conjecture
is usually false.Comment: 50 pages, to appear in Algebraic Geometr
- …