The proof of Theorem 3.1, as presented in the article needs more explanation and does require a little more work.

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(ℳ_i, L^ℳ) (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) = 2^g,

where g is the genus of X. On the other hand, we have

dimH⁰(Jac(X),O(2ϴ)) = 2^g.

In fact we have a natural isomorphism between these two vector spaces (See [1]). In [2], the meaning of the two equations

dimH⁰(ℳ₀, L²) = 2^g-1(2^g + 1)

dimH⁰(ℳ₁, L) = 2^g-1(2^g - 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 D_κ 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 [13], two vector spaces H⁰(ℳ₀, L⁴) and H⁰(ℳ₁, L²) are considered. In [15], 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 article, we shall show that Cornalba-Harris equality holds true in any characteristic -- even in char(k) = 2. That will be the last result on this problem for hyperelliptic curves.

In this paper, we continue to develop the systematic decomposition theory [18] 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 [2], which were left unresolved in [18]. 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.

We present a Zariski pair in affine complex plane consisting of two line arrangements, each of which has six lines.

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.

We study the geometry of the spacelike surfaces in Minkowski 4-space through their generic contact with lightlike hyperplanes.

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.