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We show that the M-canonical map of an n-dimensional complex projective manifold X of Kodaira dimension two is birational to an Iitaka fibration for a computable positive integer M. M depends on the index b of a general fibre F of the Iitaka fibration and on the Betti number of the canonical covering of F, In particular, M is a universal constant if the dimension n is smaller than or equal to 4.
Kodaira fibred surfaces are a remarkable example of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic
We introduce a relative version of the spherical objects of Seidel and Thomas. Define an object E in the derived category D(Z x X) to be spherical over Z if the corresponding functor from D(Z) to D(X) gives rise to autoequivalences of D(Z) and D(X) i
Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a sub
We bound the slope of sweeping curves in the fourgonal locus of the moduli space of genus g algebraic curves. Our results follow from some Bogomolov-type inequalities for weakly positive rank two vector bundles on ruled surfaces.
Motivated by the study of rationally connected fibrations (and the MRC quotient) we study different notions of birationally simple fibrations. We say a fibration of smooth projective varieties is Chow constant if pushforward induces an isomorphism on