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# Especies con potencial y superficies con puntos orbifold

Ponente: Daniel Labardini
Institución: IM-UNAM
Tipo de Evento: Investigación
Cuándo 13/03/2017 de 16:00 a 17:45 Salón 1 de seminarios vCal iCal

Let $$(\Sigma,M,O)$$ be a surface with marked points and order-2 orbifold points which is either unpunctured or once-punctured closed, and let $$\omega$$ be a function from O to {1,4}. For each triangulation $$\tau$$ of $$(\Sigma,M,O)$$ we construct a cochain complex $$C^\bullet(\tau,\omega)$$ with coefficients in the field with two elements. A colored triangulation of $$(\Sigma,M,O,\omega)$$ is defined to be a pair $$(\tau,\xi)$$ consisting of a triangulation of $$(\Sigma,M,O)$$ and a 1-cocycle of $$C^\bullet(\tau,\omega)$$; the combinatorial notion of colored flip of colored triangulations is then defined as a refinement of the notion of flip of triangulations. Our main construction associates to each colored triangulation a species and a potential, and our main result shows that colored triangulations related by a colored flip have SPs related by the corresponding SP-mutation as defined in the prequel to this paper. We define the flip graph of colored triangulations of $$(\Sigma,M,O,\omega)$$ as the graph whose vertices are the pairs $$(\tau[\xi])$$ consisting of a triangulation \tau and a cohomology class in H^1(C^\bullet(\tau,\omega)), with an edge between $$(\tau,[\xi])$$ and $$(\sigma,[\zeta])$$ whenever $$(\tau,\xi')$$ and $$(\sigma,\zeta')$$ are related by a colored flip for some cocycles $$\xi'$$ and $$\zeta'$$ respectively homologous to $$\xi$$ and $$\zeta$$; we prove that this graph is disconnected if $$\Sigma$$ is not contractible. Furthermore, we show that if $$(\tau,\xi)$$ and $$(\tau,\xi')$$ satisfy $$[\xi]=[\xi']$$, then the arising Jacobian algebras are isomorphic. We also classify the non-degenerate potentials on the species associated to a given $$(\tau,\xi)$$.
The species constructed here are species realizations of the $$2^{|O|}$$ skew-symmetrizable matrices assigned by Felikson-Shapiro-Tumarkin to any given $$\tau$$.