Abstract. Let X be a smooth projective variety of Albanese fiber dimension 1 and of general type. We prove that the translates through 0 of all components of V 0 (ωX ) generate Pic 0 (X). We then study the pluricanonical maps of X. We show that |4KX | induces a birational map of X.

Abstract. We introduce an algorithm that exploits a combinatorial symmetry of an arrangement in order to produce a geometric reflection between two disconnected components of its moduli space. We apply this method to disqualify three real examples found in previous work by the authors from being Zariski pairs. Robustness is shown by its application to complex cases, as well.

We study the Cayley-Bacharach property on complex projective smooth varieties of dimension n 2 for zero dimensional subscheme defined by the zero set of the wedge of r − n + 1 global sections of a rank r n vector bundle, and give a construction of high rank reflexive sheaves and vector bundles from codimension 2 subschemes.