arXiv subjects

Miguel Ángel Barja

Publications and source records attributed to Miguel Ángel Barja.

5 recordsLinked to original sources

arXiv2018

Higher dimensional Clifford-Severi equalities

Let $X$ be a smooth complex projective variety, $a\colon X\rightarrow A$ a morphism to an abelian variety such that $\mathrm{Pic}^0(A)$ injects into $\mathrm{Pic}^0(X)$ and let $L$ be a line bundle on $X$; denote by $h^0_a(X,L)$ the minimum of $h^0(X,L\otimes a^*α)$ for $α\in \mathrm{Pic}^0(A)$. The so-called Clifford-Severi inequalities have been proven in arXiv:1303.3045 [math.AG] and arXiv:1606.03290 [math.AG]}; in particular, for any $L$ there is a lower bound for the volume given by: $$\mathrm{vol}(L)\ge n! h^0_a(X,L),$$ and, if $K_X-L$ is pseudoeffective, $$\mathrm{vol}(L)\ge 2n! h^0_a(X,L).$$ In this paper we characterize varieties and line bundles for which the above Clifford-Severi inequalities are equalities.

math.AG↗

arXiv2018

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 étale 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 $α\in \rm{Pic}^0(A)$ general, we study the restricted linear system $|L^{(d)}\otimes a^*α|_{|T^{(d)}}$: if for some $d$ this gives a generically finite map $φ^{(d)}$, we show that f $φ^{(d)}$ is independent of $α$ or $d$ sufficiently large and divisible, and is induced by the {\em eventual map} $φ\colon T\to Z$ such that $a_{|T}$ factorizes through $φ$. The generic value $h^0_a(X_{|T}, L)$ of $h^0(X_{|T}, L\otimesα)$ 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 $C(m)={\mathcal O}(m!)$.

math.AG↗

arXiv2016

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 $χ(K_X)>0$). 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.

math.AG↗

arXiv2015

Xiao's conjecture for general fibred surfaces

We prove that the genus $g$, the relative irregularity $q_f$ and the Clifford index $c_f$ of a non-isotrivial fibration $f$ satisfy the inequality $q_f \leq g - c_f$. This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.

math.AG↗

arXiv2015

Surfaces on the Severi line

Let S be a minimal complex surface of general type and of maximal Albanese dimension; by the Severi inequality one has $K^2_S\geq 4χ(\mathcal O_S)$. We prove that the equality $K^2_S=4χ(\mathcal O_S)$ holds if and only if $q(S):= h^1(\mathcal O_S)=2$ and the canonical model of $S$ is a double cover of the Albanese surface branched on an ample divisor with at most negligible singularities.

math.AG↗