Stability and singularities of relative hypersurfaces

We study relative hypersurfaces over curves, and prove an instability condition for the fibres. This gives an upper bound on the log canonical threshold of the relative hypersurface. We compare these results with the information that can be derived from Nakayama's Zariski decomposition of effective divisors on relative projective bundles.Comment: 26 pages, 1 figure, revised version with minor change

Linear stability of projected canonical curves with applications to the slope of fibred surfaces

Let f :S\to B be a non locally trivial fibred surface. We prove a lower bound for the slope of f depending increasingly from the relative irregularity of f and the Clifford index of the general fibres.Comment: Latex, 19 pages, revised version; to appear in J. of Math. Soc. of Japa

Slope inequalities for fibred surfaces via git

In this paper we present a generalisation of a theorem due to Cornalba and Harris, which is an application of Geometric Invariant Theory to the study of invariants of fibrations. In particular, our generalisation makes it possible to treat the problem of bounding the invariants of general fibred surfaces. As a first application, we give a new proof of the slope inequality and of a bound for the invariants associated to double cover fibrations

Slopes of trigonal fibred surfaces and of higher dimensional fibrations

We give lower bounds for the slope of higher dimensional fibrations f : X \u2192 B over curves under conditions of GIT-semistability of the fibres, using a generalization of a method of Cornalba and Harris. With the same method we establish a sharp lower bound for the slope of trigonal fibrations of even genus and general Maroni invariant; in particular this result proves a conjecture due to Harris and Stankova-Frenkel

Positivity properties of relative complete intersections

We give conditions for $f$-positivity of relative complete intersections in projective bundles. We also derive an instability result for the fibres.Comment: 7 page

Stability conditions and positivity of invariants of fibrations

We study three methods that prove the positivity of a natural numerical invariant associated to $1-$parameter families of polarized varieties. All these methods involve different stability conditions. In dimension 2 we prove that there is a natural connection between them, related to a yet another stability condition, the linear stability. Finally we make some speculations and prove new results in higher dimension.Comment: Final version, to appear in the Springer volume dedicated to Klaus Hulek on the occasion of his 60-th birthda

Fibrations of Campana general type on surfaces

We construct complex surfaces with genus two fibrations over P^1 having special fibres such that the minimum of the multiplicities of the components is 65 2 whereas the g.c.d is 1. We can then produce new examples of fibred surfaces without multiple fibres which are of \u201cgeneral type\u201d according to the definition of Campana. We prove that these surfaces are of general type and simply connected; and we compute in some cases their invariants. Moreover, we extend the construction obtaining general type fibrations of any even genus on simply connected surfaces. All our examples are defined over number field