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A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence

Alessandro Chiodo, Yongbin Ruan (2011)

Annales de l’institut Fourier

We show how the Landau–Ginzburg/Calabi–Yau correspondence for the quintic three-fold can be cast into a global mirror symmetry framework. Then we draw inspiration from Berglund–Hübsch mirror duality construction to provide an analogue conjectural picture featuring all Calabi–Yau hypersurfaces within weighted projective spaces and certain quotients by finite abelian group actions.

A group action on Losev-Manin cohomological field theories

Sergey Shadrin, Dimitri Zvonkine (2011)

Annales de l’institut Fourier

We discuss an analog of the Givental group action for the space of solutions of the commutativity equation. There are equivalent formulations in terms of cohomology classes on the Losev-Manin compactifications of genus  0 moduli spaces; in terms of linear algebra in the space of Laurent series; in terms of differential operators acting on Gromov-Witten potentials; and in terms of multi-component KP tau-functions. The last approach is equivalent to the Losev-Polyubin classification that was obtained...

A limit linear series moduli scheme

Brian Osserman (2006)

Annales de l’institut Fourier

We develop a new, more functorial construction for the basic theory of limit linear series, which provides a compactification of the Eisenbud-Harris theory. In an appendix, in order to obtain the necessary dimensional lower bounds on our limit linear series scheme we develop a theory of “linked Grassmannians”; these are schemes parametrizing sub-bundles of a sequence of vector bundles, which map into one another under fixed maps of the ambient bundles.

A remark on a conjecture of Hain and Looijenga

Carel Faber (2011)

Annales de l’institut Fourier

We show that the natural generalization of a conjecture of Hain and Looijenga to the case of pointed curves holds for all g and n if and only if the tautological rings of the moduli spaces of curves with rational tails and of stable curves are Gorenstein.

A remark on the homotopical dimension of some moduli spaces of stable Riemann surfaces

Gabriele Mondello (2008)

Journal of the European Mathematical Society

Using a result of Harer, we prove certain upper bounds for the homotopical/cohomological dimension of the moduli spaces of Riemann surfaces of compact type, of Riemann surfaces with rational tails and of Riemann surfaces with at most k rational components. These bounds would follow from conjectures of Looijenga and Roth–Vakil.

A remark on the Picard group of spin moduli space

Maurizio Cornalba (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We describe a number of classes in the Picard group of spin moduli space and determine the relations they satisfy; as an application we show that the Picard group in question contains 4-torsion elements.

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