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Let ℳ0 [resp. ℳ1] be a coarse moduli space of rank 2 semistable vector bundles of even [resp. odd] degree with fixed determinant on a smooth projective curve X. The Picard group is infinite cyclic . Let L be the ample generator. The dimension of a vector space H0(ℳi, Lℳ) (i = 0, 1) is given by the Verlinde formula. For small m > 0, the meaning of this dimension can be explained in the framework of algebraic geometry. For example, we have
dimH0(ℳ0, L) = 2g,
where g is the genus of X. On the other hand, we have
dimH0(Jac(X),O(2ϴ)) = 2g.
In fact we have a natural isomorphism between these two vector spaces (See ). In , the meaning of the two equations
dimH0(ℳ0, L2) = 2g-1(2g + 1)
dimH0(ℳ1, L) = 2g-1(2g - 1)
are clarified. The above dimensions are the number of even or odd theta characterictics on X. Beauville associated to an even [resp. odd] theta characterictic κ a divisor Dκ on ℳ0 [resp. ℳ1] that can be described from a moduli-theoretic viewpoint, and proved that they form a basis of H0(ℳ0, L2) [resp. H0(ℳ1, L)]. In , two vector spaces H0(ℳ0, L4) and H0(ℳ1, L2) are considered. In , Pauly deals with a parabolic case.
The purpose of this paper is to carry out a similar study for a moduli space ℳ Par(ℙ1; I) of rank 2 semistable parabolic vector bundles with half weights of degree zero on ℙ1.
In this paper, we continue to develop the systematic decomposition theory  for the generic Newton polygon attached to a family of zeta functions over finite fields and more generally a family of L-functions of n-dimensional exponential sums over finite fields. Our aim is to establish a new collapsing decomposition theorem (Theorem 3.7) for the generic Newton polygon. A number of applications to zeta functions and L-functions are given, including the full form of the remaining 3 and 4-dimensional cases of the Adolphson-Sperber conjecture , which were left unresolved in . To make the paper more readable and useful, we have included an expanded introductory section as well as detailed examples to illustrate how to use the main theorems.
The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's G2 quantum invariants. Our method consists of the study of Markov traces on a suitable tower of quotients of cubic Hecke algebras extending Jones approach.
For every supersingular K3 surface X in characteristic 2, there exists a homogeneous polynomial G of degree 6 such that X is birational to the purely inseparable double cover of ℙ2 defined by ω2 = G. We present an algorithm to calculate from G a set of generators of the numerical Néron-Severi lattice of X. As an application, we investigate the stratification defined by the Artin invariant on a moduli space of supersingular K3 surfaces of degree 2 in characteristic 2.