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.

Singularities on demi-normal varieties

Abstract: The birational classification of varieties inevitably leads to the study of singularities. The types of singularities that occur in this context have been studied by Mori, Kollar, Reid, and others, beginning in the 1980s with the introduction of the Minimal Model Program. Normal singularities that are terminal, canonical, log terminal, and log canonical, and their non-normal counterparts, are typically studied by using a resolution of singularities (or a semi-resolution), and finding numerical conditions that relate the canonical class of the variety to that of its resolution. In order to do this, it has been assumed that a variety X is has a $${\mathbb {Q}}$$ -Cartier canonical class: some multiple $$mK_X$$ of the canonical class is Cartier. In particular, this divisor can be pulled back under a resolution $$f: Y \rightarrow X$$ by pulling back its local sections. Then one has a relation $$K_Y \sim \frac{1}{m}f^*(mK_X) + \sum a_iE_i$$ . It is then the coefficients of the exceptional divisors $$E_i$$ that determine the type of singularities that belong to X. It might be asked whether this $${\mathbb {Q}}$$ -Cartier hypothesis is necessary in studying singularities in birational classification. de Fernex and Hacon (Compos Math 145:393–414, 2009) construct a boundary divisor $$\Delta $$ for arbitrary normal varieties, the resulting divisor $$K_X + \Delta $$ being $${\mathbb {Q}}$$ -Cartier even though $$K_X$$ itself is not. This they call (for reasons that will be made clear) an m-compatible boundary for X, and they proceed to show that the singularities defined in terms of the pair $$(X,\Delta )$$ are none other than the singularities just described, when $$K_X$$ happens to be $${\mathbb {Q}}$$ -Cartier. Thus, a wider context exists within which one can study singularities of the above types. In the present paper, we extend the results of de Fernex and Hacon (Compos Math 145:393–414, 2009) still further, to include demi-normal varieties without a $${\mathbb {Q}}$$ -Cartier canonical class. Our main result is that m-compatible boundaries exist for demi-normal varieties (Theorem 1.1). This theorem provides a link between the theory of singularities for arbitrary demi-normal varieites (whose canonical class may not $$\hbox {be } {\mathbb {Q}}$$ -Cartier), that theory being developed in the present paper, and the established theory of singularities of pairs.

Newton-Okounkov bodies and Picard numbers on surfaces

Abstract: We study the shapes of all Newton-Okounkov bodies $\Delta_{v}(D)$ of a given big divisor $D$ on a surface $S$ with respect to all rank 2 valuations $v$ of $K(S)$. We obtain upper bounds for, and in many cases we determine exactly, the possible numbers of vertices of the bodies $\Delta_{v}(D)$. The upper bounds are expressed in terms of Picard numbers and they are birationally invariant, as they do not depend on the model $\tilde{S}$ where the valuation $v$ becomes a flag valuation. We also conjecture that the set of all Newton-Okounkov bodies of a single ample divisor $D$ determines the Picard number of $S$, and prove that this is the case for Picard number 1, by an explicit characterization of surfaces of Picard number 1 in terms of Newton-Okounkov bodies.

Determination of Certain t-Groups

Abstract: Transiso graph $$\varGamma _d(G)$$ is a graph defined for a finite group G and a divisor d of the order of G. Subgroups of G are vertices of $$\varGamma _d(G)$$ and two subgroups are said to be connected by an edge if they have a pair of isomorphic normalized right transversals. A group G is called a t-group if graph $$\varGamma _d(G)$$ is a complete graph for each divisor d of |G|. Completeness of transiso graphs for some groups like abelian groups, p-groups of order up to $$p^5$$ where p is an odd prime, dihedral groups and dicyclic groups etc. are already discussed in the literature. In the present research article, we have discussed the completeness of transiso graphs for 2-groups of the order $$2^5$$ and concluded that there are only two non-abelian t-groups (precisely dihedral group and generalized quaternion group) of the order $$2^5$$ .

Goldschmidt division implementation method based on divisor mapping

Abstract: The invention discloses a Goldschmidt division implementation method based on divisor mapping. The method includes that a floating point form dividend Nf and a floating point form divisor Df are firstly standardized in a f* 2e form, the standardized dividend is denoted with N and the standardized divisor is denoted with D; a boundary value p is obtained according to a given minimum relative error E and iterations M; when the standardized divisor D is in an interval [1, p], M iterations are directly conducted; when the standardized divisor D is in an interval [p, 2], D is mapped in the interval [1, p], and then the M iterations are conducted. When the iterations are conducted, an initial value F0=2-D0. A division result of f part is obtained through M iterations, the division result of the f part is finally combined with a subtraction result of 2e part, and then a final division operation result is obtained. The Goldschmidt division implementation method based on the divisor mapping does not need an initial estimated value, and thereby a mass of storage resources are saved.