Open Access. Powered by Scholars. Published by Universities.®
Physical Sciences and Mathematics Commons™
Open Access. Powered by Scholars. Published by Universities.®
Articles 1 - 4 of 4
Full-Text Articles in Physical Sciences and Mathematics
Mirror Symmetry For Log Calabi-Yau Surfaces I, Mark Gross, Paul Hacking, Sean Keel
Mirror Symmetry For Log Calabi-Yau Surfaces I, Mark Gross, Paul Hacking, Sean Keel
Paul Hacking
We give a cononical sythetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family is constructed as the spectrum of an explicit algebra structure on a vector space with canonical basis and multiplication rule defined in terms of counts of rational curves on Y meeting D in a single point. The elements of the canonical basis are called theta functions. Their construction depends crucially on the Gromov-Witten theory of the pair (Y,D)
Birational Geometry Of Cluster Algebras, Mark Gross, Paul Hacking, Sean Keel
Birational Geometry Of Cluster Algebras, Mark Gross, Paul Hacking, Sean Keel
Paul Hacking
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 [Spe13] of upper cluster algebras which are not finitely generated, and show that the Fock-Goncharov dual basis conjecture is usually false.
Canonical Bases For Cluster Algebras, Mark Gross, Paul Hacking, Sean Keel, Maxim Kontesevich
Canonical Bases For Cluster Algebras, Mark Gross, Paul Hacking, Sean Keel, Maxim Kontesevich
Paul Hacking
In GHK11, Conjecture 0.6, the first three authors conjectured the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture, FG06, Conjecture 4.3. In particular, under …
Moduli Of Surfaces With An Anti-Canonical Cycle, Mark Gross, Paul Hacking, Sean Keel
Moduli Of Surfaces With An Anti-Canonical Cycle, Mark Gross, Paul Hacking, Sean Keel
Paul Hacking
We prove a global torelli theorem for pairs (Y,D) where Y is a smooth projective rational surface and D ∈ |−Ky | is a cycle of rational curves, as conjectured by Friedman in 1984. In addition, we construct natural universal families for such pairs.