On Fano varieties whose effective divisors are numerically eventually free

Threefolds whose canonical bundles are not numerically effective.

Let X be an arbitrary nonsingular projective 3-fold whose canonical bundle is not numerically effective. Then we have: (i) X contains an exceptional divisor of several types, which we classify explicitly, (ii) X has a morphism to a projective nonsingular surface whose fibers are conics, (iii) X has a morphism to a projective nonsingular curve whose general fibers are Del Pezzo surfaces, or (iv)...

Effective divisors on moduli spaces of curves and abelian varieties

The pseudo-effective cone Eff(X) of a smooth projective variety X is a fundamental, yet elusive invariant. On one hand, a few general facts are known: the interior of the effective cone is the cone of big divisors so, in particular, X is of general type if and only if KX ∈ int(Eff(X)); less obviously [4], a variety X is uniruled if and only if KX is not pseudo-effective and the dual of Eff(X) i...

THEOREMS ABOUT GOOD DIVISORS ON LOG FANO VARIETIES ( CASE OF INDEX r > n - 2 )

Canonical Divisors on T-varieties

Generalising toric geometry we study compact varieties admitting lower dimensional torus actions. In particular we describe divisors on them in terms of convex geometry and give a criterion for their ampleness. These results may be used to study Fano varieties with small torus actions. As a first result we classify log del Pezzo C∗-surfaces of Picard number 1 and Gorenstein index ≤ 3. In furthe...

Solvable Group Representations and Free Divisors Whose Complements Are K(π, 1)‘s

A classical result of Arnold and Brieskorn [Bk], [Bk2] states that the complement of the discriminant of the versal unfolding of a simple hypersurface singularity is a K(π, 1). Deligne [Dg] showed this result could be placed in the general framework by proving that the complement of an arrangement of reflecting hyperplanes for a Coxeter group is again a K(π, 1) (and more generally for simplicia...