# Morphisms and faces of pseudo‐effective cones

@article{Fulger2016MorphismsAF, title={Morphisms and faces of pseudo‐effective cones}, author={Mihai Fulger and Brian Lehmann}, journal={Proceedings of the London Mathematical Society}, year={2016}, volume={112} }

Let π:X→Y be a morphism of projective varieties and suppose that α is a pseudo‐effective numerical cycle class satisfying π*α=0 . A conjecture of Debarre, Jiang, and Voisin predicts that α is a limit of classes of effective cycles contracted by π . We establish new cases of the conjecture for higher codimension cycles. In particular, we prove a strong version when X is a fourfold and π has relative dimension 1.

## 16 Citations

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We exhibit infinitely many extremal effective codimension-$k$ cycles in $\overline{\mathcal{M}}_{g,n}$ in the cases
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## References

SHOWING 1-10 OF 20 REFERENCES

### Pseudo-effective classes and pushforwards

- Mathematics
- 2013

Given a morphism between complex projective varieties, we make several conjectures on the relations between the set of pseudo-effective (co)homology classes which are annihilated by pushforward and…

### Kernels of numerical pushforwards

- Mathematics
- 2014

Abstract Let π : X → Y be a morphism of projective varieties and π∗ :Nk(X) → Nk(Y) the pushforward map of numerical cycle classes. We show that when the Chow groups of points of the fibers are as…

### Positive cones of dual cycle classes

- Mathematics
- 2014

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…

### The pseudo-effective cone of a compact K\

- Mathematics
- 2004

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a…

### Pseudoeffective and nef classes on abelian varieties

- MathematicsCompositio Mathematica
- 2011

Abstract We study the cones of pseudoeffective and nef cycles of higher codimension on the self product of an elliptic curve with complex multiplication, and on the product of a very general abelian…

### Geometric characterizations of big cycles

- Mathematics
- 2013

A numerical equivalence class of k-cycles is said to be big if it lies in the interior of the closed cone generated by effective classes. We develop several geometric criteria that distinguish big…

### Varieties with generically nef tangent bundles

- Mathematics
- 2008

We study the geometry of projective manifolds whose tangent bundles are nef on sufficiently general curves (i.e. the tangent bundle is generically nef) and show that manifolds whose anticanonical…

### Zariski decompositions of numerical cycle classes

- Mathematics
- 2013

We construct a Zariski decomposition for cycle classes of arbitrary codimension. This decomposition is an analogue of well-known constructions for divisors. Examples illustrate how Zariski…

### Extremal higher codimension cycles on moduli spaces of curves

- Mathematics
- 2014

We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space M¯g,n of stable genus g curves with n ordered marked points. In particular,…