Brian Lehmann

Positive cones of dual cycle classes

We study generalizations for higher codimension cycles of several well-known definitions of the nef cone of divisors on a projective variety. These generalizations fix some of the pathologies exhibited by the classical nef cone of higher codimension classes. As an application, we recover the expected properties of the pseudoeffective cones

Reduction maps and minimal model theory

\overline{Eff}_{k}(X)

A cone theorem for nef curves

for all k.

On Eckl’s pseudo-effective reduction map

We use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a klt pair

Convexity and Zariski decomposition structure

(X,\Delta)

Kernels of numerical pushforwards

can be detected on the base of the

On the geometry of thin exceptional sets in Manin’s conjecture

(K_{X}+\Delta)

Morphisms and faces of pseudo-effective cones

-trivial reduction map. Thus we show that the main conjectures of the minimal model program can be interpreted as a natural statement on the existence of curves on

Comparing numerical dimensions

X

Zariski decompositions of numerical cycle classes

.