Slopes of trigonal fibred surfaces and of higher dimensional fibrations

We give lower bounds for the slope of higher dimensional fibrations 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.Comment: 11 page

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

The eventual paracanonical map of a variety of maximal Albanese dimension

Let $X$ be a smooth complex projective variety such that the Albanese map of $X$ is generically finite onto its image. Here we study the so-called eventual $m$-paracanonical map of $X$ (when $m=1$ we also assume $\chi(K_X)>0$). We show that for $m=1$ this map behaves in a similar way to the canonical map of a surface of general type, while it is birational for $m>1$. We also describe it explicitly in several examples.Comment: 13 page

Linear systems on irregular varieties

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety, $a\colon X\rightarrow A$ a morphism to an abelian variety such that $\rm{Pic}^0(A)$ injects into $\rm{Pic}^0(T)$ and let $L$ be a line bundle on $X$. Denote by $X^{(d)}\to X$ the connected \'etale cover induced by the $d$-th multiplication map of $A$, by $T^{(d)} \subseteq X^{(d)}$ the preimage of $T$ and by $L^{(d)}$ the pull-back of $L$ to $X^{(d)}$. For $\alpha\in \rm{Pic}^0(A)$ general, we study the restricted linear system $|L^{(d)}\otimes a^*\alpha|_{|T^{(d)}}$: if for some $d$ this gives a generically finite map $\varphi^{(d)}$, we show that f $\varphi^{(d)}$ is independent of $\alpha$ or $d$ sufficiently large and divisible, and is induced by the {\em eventual map} $\varphi\colon T\to Z$ such that $a_{|T}$ factorizes through $\varphi$. The generic value $h^0_a(X_{|T}, L)$ of $h^0(X_{|T}, L\otimes\alpha)$ is called the {\em (restricted) continuous rank.} We prove that if $M$ is the pull back of an ample divisor of $A$, then $x\mapsto h^0_a(X_{|T}, L+xM)$ extends to a continuous function of $x\in\mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly. In the case when $X$ and $T$ are smooth, combining the above results we prove Clifford-Severi type inequalities, i.e., geographical bounds of the form $$\rm{vol}_{X|T}(L)\geq C(m) h^0_a(X_{|T},L),$$ where $C(m)={\mathcal O}(m!)$.Comment: Revised version, 37 pages. The final section has been remove

Galois closure and Lagrangian varieties

We use Galois closures of finite rational maps between complex projective varieties to introduce a new method for producing varieties such that the holomorphic part of the cup product map has non-trivial kernel. We then apply our result to the two-dimensional case and we construct a new family of surfaces which are Lagrangian in their Albanese variety. Moreover, we analyze these surfaces computing their Chern invariants, and proving that they are not fibred over curves of genus greater than one.Comment: 36 pages, 3 figure

On the rank of the flat unitary summand of the Hodge bundle

Let $f\colon S\to B$ be a non-isotrivial fibred surface. We prove that the genus $g$, the rank $u_f$ of the unitary summand of the Hodge bundle $f_*\omega_f$ and the Clifford index $c_f$ satisfy the inequality $u_f \leq g - c_f$. Moreover, we prove that if the general fibre is a plane curve of degree $\geq 5$ then the stronger bound $u_f \leq g - c_f-1$ holds. In particular, this provides a strengthening of the bounds of \cite{BGN} and of \cite{FNP}. The strongholds of our arguments are the deformation techniques developed by the first author in \cite{Rigid} and by the third author and Pirola in \cite{PT}, which display here naturally their power and depht.Comment: 19 pages, revised versio

The slope of fibred surfaces: unitary rank and Clifford index

We prove new slope inequalities for relatively minimal fibred surfaces, showing an influence of the relative irregularity, of the unitary rank and of the Clifford index on the slope. The argument uses Xiao's method and a new Clifford-type inequality for subcanonical systems on non-hyperelliptic curves.Comment: 23 page

Linear series on curves: stability and Clifford index

We study concepts of stabilities associated to a smooth complex curve together with a linear series on it. In particular we investigate the relation between stability of the associated Dual Span Bundle and linear stability. Our result implies a stability condition related to the Clifford index of the curve. Furthermore, in some of the cases, we prove that a stronger stability holds: cohomological stability. Eventually using our results we obtain stable vector bundles of integral slope 3, and prove that they admit theta-divisors.Comment: 24 page

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