arXiv subjects
Publications and source records attributed to A. G. Zamora.
Let $L$ be a globally generated line bundle over a smooth irreducible complex projective surface $X$. The syzygy bundle $M_{L}$ is the kernel of the evaluation map $H^0(L)\otimes\mathcal O_X\to L$. We prove the $L$-stability of $M_L$ for Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. The $(-K_X)$-stability of syzygy bundles $M_L$ over del Pezzo surfaces is also obtained.
We extend the concept of Segre's Invariant to vector bundles on a surface $X$. For $X=\mathbb{P}^2$ we determine what numbers can appear as the Segre Invariant of a rank $2$ vector bundle with given Chern's classes. The irreducibility of strata with fixed Segre's invariant is proved and its dimensions are computed. Finally, we present applications to the Brill-Noether's Theory for rank $2$ vector bundles on $\mathbb{P}^2.$
Given a semistable non-isotrivial fibered surface $f:X\to \mathbb{P}^1$ it was conjectured by Tan and Tu that if $X$ is of general type, then $f$ admits at least $7$ singular fibers. In this paper we prove this conjecture in several particular cases, i.e. assuming $f$ is obtained from blowing-up the base locus of a transversal pencil on an exceptional minimal surface $S$ or assuming that $f$ is obtained as the blow-up of the base locus of a transversal and adjoint pencil on a minimal surface.
The Poincare Problem can be reduced to a problem on fibered surfaces, concretely, to bound the genus of the fibration by means of numerical information of the canonical sheaf of the associated foliation. In this paper we: 1. explain how this reduction can be carried out, 2. apply unipotent reduction to obtain some possible answers to the problem, 3. obtain an answer under some hypotheses on the eigenvalues of the singularities of the foliation