Stability and singularities of relative hypersurfaces

We study relative hypersurfaces, and prove an instability condition for the fibres. This is the starting point for an investigation of the geometry of effective divisors on relative projective bundles.Preprin

Numerical Bounds of Canonical Varieties

We study surfaces and threefolds whose canonical maps are birational (canonical surfaces and threefolds). In the case of canonical surfaces we prove that the known inequlity for its invariants is not sharp if the surface is irregular. In the case of canonical threefolds we give a new lower bound for the self-intersection of the canonical divisor, depending on the geometric genus and the irregularity of the threefold

Slope inequalities for fibrations of non-maximal Albanese dimension

The version of record is available online at: http://dx.doi.org/10.1007/s40574-021-00294-5We study and obtain Slope inequalities for fibred irregular varieties of non-maximal Albanese dimension. We give a comparison theorem between Clifford–Severi and Slope inequalities for this type of fibrations. We also obtain a set of Slope inequalities considering the geometry of the Albanese map and the associated eventual maps.The author was supported by MINECO PID2019-103849GB-I00 “Geometría Álgebra, Topología y aplicaciones multidisciplinares” and by Generalitat de Catalunya SGR2017-932Peer ReviewedPostprint (author's final draft

Positivity properties of relative complete intersections

PreprintWe give conditions for f-positivity of relative complete intersections in projective bundles. We also derive an instability result for the fibres.Preprin

On the slope of fibred surfaces

We give an asymptotically sharp lower bound for the slope $\lambda (f)$ of a fibration $f:S\longrightarrow B$, where $S$ is a surface and $B$ is a curve, if there exists an involution on the general fibre $F$ of $f$. We also construct a new lower bound of $\lambda (f)$ depending increasingly on the irregularity of $S$; as an application of this new bound we have a criteria to control the existence of other fibrations on $S$