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Combinatorial aspects of toric varieties

Christine von Renesse, University of Massachusetts Amherst


According to Batyrev the Mori cone of a smooth, complete and projective toric variety can be generated by primitive relations. A primitive relation comes from a primitive collection, which is a set of 1-dimensional cones of the fan Σ such that the whole collection {ρ1, ..., ρk} does not generate a cone in Σ, but every subset does. To prove Batyrev's smooth case you can relate wall collections and primitive collections. I generalize Batyrev's statement to the non-complete, non-smooth but simplicial case and to the non-simplicial case. ^ Lawrence toric varieties arise as GIT-quotients. Hausel and Sturmfels showed that the cohomology of Lawrence toric varieties is independent of the GIT parameter. I will give a different proof for this result. Moreover, I will show that a natural way of generalizing these varieties does not have independent cohomology anymore by presenting some counter-examples. ^

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Recommended Citation

von Renesse, Christine, "Combinatorial aspects of toric varieties" (2007). Doctoral Dissertations Available from Proquest. AAI3289222.