Kalle Karu

Professor

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2021)
Negative curves in blowups of weighted projective planes (2020)

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 isa Mori dream space if and only if it contains two irreducible disjoint curves; one of themnecessarily having non-positive self-intersection. We call such a curve a “negative curve”. Asignificant part of this thesis is dedicated to the study of such negative curves, as they largelygovern the Mori dream space property for these varieties.Our study begins by constructing two one-parameter families of negative curves andsubsequently a larger two-parameter class of negative curves having the previous twofamilies as boundary cases.Once such a variety is known to contain a negative curve, we determine if it contains adisjoint curve by using different procedures. For example, prime characteristic and coho-mological methods. Furthermore, we introduce an independent technique that applies to abroader class of cases. As a result, for each of the negative curves constructed we provideexamples and non-examples of Mori dream spaces containing the curve.

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Master's Student Supervision (2010 - 2020)
Moduli Space of Sheaves on Fans (2011)

A conjecture of H. Hopf states that if x(M²n) is a closed, Riemannian manifold of nonpositive sectional curvature, then its Euler characteristic x(M²n), should satify (-1)n x(M²n)≥ 0. Ruth Charney and Michael Davis investigated the conjecture in the context of piecewise Euclidean manifolds having "nonpositive curvature" in the sense of Gromov's CAT(0) inequality. In that context the conjecture can be reduced to a local version which predicts the sign of a "local Euler characteristic" at each vertex. They stated precisely various conjectures in their paper which we are interested in one of them stated as Conjecture D (see [1]) which is equivalent to the Hopf Conjecture for piecewise Euclidean manifolds cellulated by cubes.The goal of this thesis is to study the Charney - Davis Conjecture stated as Conjecture (D) by using sheaves on fans.

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