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In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [G. Banaszak and P. Krasoń, On a local to global principle in étale K-groups of curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183–201], G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for étale K -theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases, this result is the best possible i.e. if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for étale K -theory of a curve. The dynamical local to global principle for the groups of Mordell–Weil type has recently been considered by S. Barańczuk in [S. Barańczuk, On a dynamical local-global principle in Mordell-Weil type groups, Expo. Math. 35(2) (2017) 206–211]. We show that all our results remain valid for Quillen K -theory of if the Bass and Quillen–Lichtenbaum conjectures hold true for . [ABSTRACT FROM AUTHOR]

In this paper, we first establish a degeneracy second main theorem for algebraic curves from compact complex Riemann surfaces into projective varieties intersecting hypersurfaces in subgeneral position. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in ℝ m with finite total curvature in the case where the Gauss map may be algebraic degenerate. Our results generalize and improve the previous results in the field. [ABSTRACT FROM AUTHOR]

Let (E , V) be a general generated coherent system of type (n , d , n + m) on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of E to the semistability of the kernel of the evaluation map V ⊗ 𝒪 X → E. The aim of this paper is to obtain results on the existence of generated coherent systems and use them to prove Butler's Conjecture in some cases. The strongest results are obtained for type (2 , d , 4) , which is the first previously unknown case. [ABSTRACT FROM AUTHOR]

An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a maximal isotropic subbundle. This invariant and the induced stratifications on moduli spaces of orthogonal bundles were studied for bundles of even rank in [I. Choe and G. H. Hitching, A stratification on the moduli space of symplectic and orthogonal bundles over a curve, Internat. J. Math. 25(5) (2014), Article ID: 1450047, 27pp.]. In this paper, we obtain analogous results for bundles of odd rank. We compute the sharp upper bound on the isotropic Segre invariant. Also we show the irreducibility of the induced strata on the moduli spaces of orthogonal bundles of odd rank, and compute their dimensions. [ABSTRACT FROM AUTHOR]

We compute the presentations of fundamental groups of the complements of a class of rational cuspidal projective plane curves classified by Flenner, Zaidenberg, Fenske and Saito. We use the Zariski-Van Kampen algorithm and exploit the Cremona transformations used in the construction of these curves. We simplify and study the group presentations so obtained and determine if they are abelian, finite or big, i.e. if they contain free non-abelian subgroups. We also study the quotients of these groups to some extent. [ABSTRACT FROM AUTHOR]