UBC Theses and Dissertations

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UBC Theses and Dissertations

Negative curves in blowups of weighted projective planes González Anaya, Javier


We study the Mori dream space property for blowups at a general point of weighted projec- tive planes or, more generally, of toric surfaces with Picard number one. Such a variety is a Mori dream space if and only if it contains two irreducible disjoint curves; one of them necessarily having non-positive self-intersection. We call such a curve a “negative curve”. A significant part of this thesis is dedicated to the study of such negative curves, as they largely govern the Mori dream space property for these varieties. Our study begins by constructing two one-parameter families of negative curves and subsequently a larger two-parameter class of negative curves having the previous two families as boundary cases. Once such a variety is known to contain a negative curve, we determine if it contains a disjoint curve by using different procedures. For example, prime characteristic and coho- mological methods. Furthermore, we introduce an independent technique that applies to a broader class of cases. As a result, for each of the negative curves constructed we provide examples and non-examples of Mori dream spaces containing the curve.

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