Topic
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.
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04 Jun 1985
TL;DR: The conditions that the divisor must satisfy to have the quotient digit qi+1 predicted while computing R(i-1) are determined.
Abstract: A division algorithm with a simple selection of quotient digits including prediction is possible if the divisor is restricted to a suitable range. The conditions that the divisor must satisfy to have the quotient digit q i+1 predicted while computing R i+1 are determined. Some implementation considerations are also given.
41 citations
Posted Content•
TL;DR: For convex lattice polytopes, the question of whether the canonical map is surjective or not has been studied in the context of toric geometry as discussed by the authors, where the authors explore various variations on the question in terms of the projective toric variety.
Abstract: This paper was submitted to the Oberwolfach Conference "Combinatorial Convexity and Algebraic Geometry", October 1997.
Let $M={\mathbb Z}^r$. For convex lattice polytopes $P,P'$ in ${\mathbb R}^r$, when is $(M \cap P)+ (M \cap P') = M \cap (P + P')$?
Without any additional condition, the equality obviously does not hold.
When the pair $(M,P)$ corresponds to a complex projective toric variety $X$ and an ample divisor $D$ on $X$, it is reasonable to assume that $P'$ corresponds to an ample (or, more generally, a nef) divisor $D'$ on the same $X$. Then the question correspons to the surjectivity of the canonical map \[ H^0(X,{\mathcal O}_X(D))\otimes H^0(X,{\mathcal O}_X(D'))\to H^0(X,{\mathcal O}_X(D+D')).\]
When $X$ is nonsingular, the map is hoped to be surjective, but this remains to be an open question after more than ten years.
The paper explores various variations on the question in terms of toric geometry.
41 citations
Posted Content•
TL;DR: In this article, it was shown that every geometrically reduced projective variety of pure dimension n over a field of positive characteristic admits a morphism to projective n-space, etale away from the hyperplane H at infinity, which maps a chosen divisor into H and some chosen smooth points not on the divisors to points not in H.
Abstract: We prove that every geometrically reduced projective variety of pure dimension n over a field of positive characteristic admits a morphism to projective n-space, etale away from the hyperplane H at infinity, which maps a chosen divisor into H and some chosen smooth points not on the divisor to points not in H. This improves our earlier result in math.AG/0207150, which was restricted to infinite perfect fields. We also prove a related result that controls the behavior of divisors through the chosen point.
41 citations
TL;DR: In this article, the Fourier coefficients of generalized modular forms were studied and the authors established two Theorems asserting that $f(\tau)$ is constant if $k = 0, $f(n)$ has an empty divisor, and the coefficients have certain rationality properties.
Abstract: We study the Fourier coefficients of generalized modular forms $f(\tau)$ of integral weight $k$ on subgroups $\Gamma$ of finite index in the modular group. We establish two Theorems asserting that $f(\tau)$ is constant if $k = 0$, $f(\tau)$ has empty divisor, and the Fourier coefficients have certain rationality properties. (The result is false if the rationality assumptions are dropped.) These results are applied to the case that $f(\tau)$ has a cuspidal divisor, $k$ is arbitrary, and $\Gamma = \Gamma_{0}(N)$, where we show that $f(\tau)$ is modular, indeed an eta-quotient, under natural rationality assumptions on the Fourier coefficients. We also explain how these results apply to the theory of orbifold vertex operator algebras.
40 citations
TL;DR: In this article, it was shown that a fiber space with non-normal fibers is uniruled and that general fibers of Mori fiber spaces are rationally chain connected, and a weakening of the cone theorem for surfaces and threefolds defined over an imperfect field was obtained.
Abstract: Let $k$ be an imperfect field. Let $X$ be a regular variety over $k$ and set $Y$ to be the normalization of $(X \times_k k^{1/p^{\infty}})_{{\rm red}}$. In this paper, we show that $K_Y+C=f^*K_X$ for some effective divisor $C$ on $Y$. We obtain the following three applications. First, we show that a $K_X$-trivial fiber space with non-normal fibers is uniruled. Second, we prove that general fibers of Mori fiber spaces are rationally chain connected. Third, we obtain a weakening of the cone theorem for surfaces and threefolds defined over an imperfect field.
40 citations