Let X be a variety over an algebraically closed field, η : Ω X → Ļ a onedimensional singular foliation, and C ⊆ X a projective leaf of η. We prove that 2pa(C)−2 = deg(Ļ|C)+λ(C)−deg(C∩S) where pa(C) is the arithmetic genus, where λ(C) is the colength in the dualizing sheaf of the subsheaf generated by the Kähler differentials, and where S is the singular locus of η. We bound λ(C) and deg(C ∩ S),...

0. The Results. Let f : X → B be a fibration of a compact smooth algebraic surface over a compact Riemann surface B, denote by g ≥ 2 the genus of a generic fiber of f and by q the genus of B. Let s be the number of singular fibers of f and ωX/B be the relative dualizing sheaf. Let C1, · · · , Cn be n mutually disjoint sections of f , and denote by D the divisor ∑n j=1Cj. Then the main result we...

Given a graph covering, there is an induced surjection of critical groups. We will try to determine the structure of the kernel and give results at different levels of generality. A main tool will be dualizing the short exact sequence and working instead with the cokernel. In the case of signed graphs and two-sheeted coverings, we will show that the kernel of the induced surjection can be descr...

Let X be a complex, smooth, complete and connected curve and G be a complex simple and simply connected algebraic group. We compute the Picard group of the stack of quasi-parabolic G-bundles over X, describe explicitly its generators in case for classical G and G 2 and then identify the corresponding spaces of global sections with the vacua spaces of Tsuchiya, Ueno and Yamada. The method uses t...

A bstract Seiberg-like dualities in 2 + 1d quiver gauge theories with 4 supercharges are investigated. We consider quivers made of various combinations classical groups U( N ), Sp( SO( ) and SU( ). Our main focus is the mapping supersymmetric monopole operators across dual theories. There a simple general rule that encodes monopoles upon dualizing single node. This dictates all which not dresse...

In [3] D. Eisenbud and J. Harris posed the following question: What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type? We answer their question for one-dimensional families of smooth curves degenerating to stable curves with just two components meeting at points in general position. In this note we treat only those families whose total ...