Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Effective divisor classes on a ruled surface

Abstract: The Neron-Severi group of divisor classes modulo algebraic equivalence on a smooth algebraic surface is often not difficult to calculate, and has classically been studied as one of the fundamental invariants of the surface. A more difficult problem is the determination of those divisor classes which can be represented by effective divisors; these divisor classes form a monoid contained in the Neron-Severi group. Despite the finite generation of the whole Neron-Severi group, the monoid of effective divisor classes may or may not be finitely generated, and the methods used to explicitly calculate this monoid seem to vary widely as one proceeds from one type of surface to another in the standard classification scheme (see Rosoff, 1980, 1981). In this paper we shall use concrete vector bundle techniques to describe the monoid of effective divisor classes modulo algebraic equivalence on a complex ruled surface over a given base curve. We will find that, over a base curve of genus 0, the monoid of effective divisor classes is very simple, having two generators (which is perhaps to be expected), while for a ruled surface over a curve of genus 1, the monoid is more complicated, having either two or three generators. Over a base curve of genus 2 or greater, we will give necessary and sufficient conditions for a ruled surface to have its monoid of effective divisor classes finitely generated; these conditions point to the existence of many ruled surfaces over curves of higher genus for which finite generation fails.

Multiplicative character sums and non linear geometric codes

Abstract: Let q be a power of a prime number, Fq the finite field with q elements, n an integer dividing q−1, n≥2, and χ a character of order n of the multiplicative group F*q. If X is an algebraic curve defined over Fq and if G is a divisor on X, we define a non linear code Γ(q, X, G, n, χ) on an alphabet with n+1 letters. We compute the parameters of this code, through the consideration of some character sums.

Structures for pairs of mock modular forms with the Zagier duality

Abstract: Zagier introduced special bases for weakly holomorphic modular forms to give the new proof of Borcherds' theorem on the infinite product expansions of integer weight modular forms on $\SL_2(\ZZ)$ with a Heegner divisor. These good bases appear in pairs, and they satisfy a striking duality, which is now called the Zagier duality. After the result of Zagier, this type duality was studied broadly in various view points including the theory of a mock modular form. In this paper, we consider this problem with the Eichler cohomology theory, especially the supplementary function theory developed by Knopp. Using holomorphic Poincare series and their supplementary functions, we construct a pair of families of vector-valued harmonic weak Maass forms satisfying the Zagier duality with integer weights $-k$ and $k+2$ respectively, $k>0$, for a $H$-group. We also investigate the structures of them such as the images under the differential operators $D^{k+1}$ and $\xi_{-k}$ and quadric relations of the critical values of their $L$-functions.

Apparatus for multiple-divisor prescaler

Abstract: Disclosed is an apparatus for multiple-divisor prescaler, which includes an odd/even core divider, a divisor control logic unit, an odd number inserted mechanism, and an n-order divided-by-2 divider with changeable trigger edges. This invention uses a clock toggle mechanism to vary the trigger edges of each divided-by-2 divider in the n-order divider, and associates the odd/even core divider to realize the multiple-divisor prescaler apparatus. Thereby, it achieves the purpose of being divided by 30/31. In addition, it increases the divisor range up to 2 n−1 +2 and 2 n +1 through the use of the clock toggle mechanism.

Periodic solutions of the sinh-Gordon equation and integrable systems

Abstract: We study the space of periodic solutions of the elliptic $\sinh$-Gordon equation by means of spectral data consisting of a Riemann surface $Y$ and a divisor $D$. We show that the space $M_g^{\mathbf{p}}$ of real periodic finite type solutions with fixed period $\mathbf{p}$ can be considered as a completely integrable system $(M_g^{\mathbf{p}},\Omega,H_2)$ with a symplectic form $\Omega$ and a series of commuting Hamiltonians $(H_n)_{n \in \mathbb{N}}$. In particular we relate the gradients of these Hamiltonians to the Jacobi fields $(\omega_n)_{n\in \mathbb{N}_0}$ from the Pinkall-Sterling iteration. Moreover, a connection between the symplectic form $\Omega$ and Serre duality is established.