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.

On Resolving Singularities

Abstract: Let V be an irreducible affine algebraic variety over a field k of characteristic zero, and let (f_0,...,f_m) be a sequence of elements of the coordinate ring. There is probably no elementary condition on the f_i and their derivatives which determines whether the blowup of V along (f_0,...,f_m) is nonsingular. The result is that there indeed is such an elementary condition, involving the first and second derivatives of the $f_i,$ provided we admit certain singular blowups, all of which can be resolved by an additional Nash blowup. There is is a particular explicit sequence of ideals R=J_0, J_1, J_2,... \subset R so that V_i=Bl_{J_i}V is the i'th Nash blowup of V, with J_i|J_{i+1} for all i. Applying our earlier paper, V_i is nonsingular if and only if the ideal class of J_{i+1} divides some power of the ideal class of J_i. The present paper brings things down to earth considerably: such a divisibility of ideal classes implies that for some N\ge r+2 J_i^{N-r-2}J_{i+1}^{r+3}=J_i^NJ_{i+2}. Yet note that this identity in turn implies J_{i+2} is a divisor of some power of J_{i+1}. Thus although $V_i$ may fail to be nonsingular, when the identity holds the {\it next} variety V_{i+1} must be nonsingular. Thus the Nash question is equivalent to the assertion that the identity above holds for some sufficiently large i and N.

A divider circuit arrangement and a dual branch receiver having such a divider circuit arrangement

Abstract: A divider circuit arrangement in which in order to avoid dividing by zero the divisor (V d ) is modified by the addition of an extra signal (X a ) to form a modified divisor V′ d = V d + X a and the dividend (V i ) is modified by the addition of the product of the quotient (V o ) and the extra signal (X a ) to form a modified dividend V′ i = V i + V o X a . A particular but not exclusive application of this divider circuit arrangement is in normalising an output signal from a dual branch receiver (not shown).

Gosset Polytopes in Picard groups of del Pezzo Surfaces

Abstract: In this article, we research on the correspondences between the geometry of del Pezzo surfaces S_{r} and the geometry of Gosset polytopes (r-4)_{21}. We construct Gosset polytopes (r-4)_{21} in Pic S_{r}; Q whose vertices are lines, and we identify divisor classes in Pic S_{r} corresponding to (a-1)-simplexes, (r-1)-simplexes and (r-1)-crosspolytopes of the polytope (r-4)_{21}. Then we explain these classes correspond to skew a-lines, exceptional systems and rulings, respectively. As an application, we work on the monoidal transform for lines to study the local geometry of the polytope (r-4)_{21}. And we show Gieser transformation and Bertini transformation induce a symmetry of polytopes 3_{21} and 4_{21}, respectively.

Divisor functions and the number of sum systems

Abstract: Divisor functions have attracted the attention of number theorists from Dirichletto the present day. Here we consider associated divisor functionsc(r)j(n) which fornon-negative integersj, rcount the number of ways of representingnas an orderedproduct ofj+rfactors, of which the firstjmust be non-trivial, and their naturalextension to negative integersr.We give recurrence properties and explicit formulaefor these novel arithmetic functions. Specifically, the functionsc(−j)j(n) count, upto a sign, the number of ordered factorisations ofnintojsquare-free non-trivialfactors. These functions are related to a modified version of the M obius functionand turn out to play a central role in counting the number of sum systems of givendimensions.Sum systems are finite collections of finite sets of non-negative integers, of pre-scribed cardinalities, such that their set sum generates consecutive integers with-out repetitions. Using a recently established bijection between sumsystems andjoint ordered factorisations of their component set cardinalities,we prove a for-mula expressing the number of different sum systems in terms of associated divisorfunctions.

Note on quasi-polarized canonical Calabi-Yau threefolds

Abstract: Let $(X,L)$ be a quasi-polarized canonical Calabi-Yau threefold. In this note, we show that $\vert mL\vert$ is basepoint free for $m\geq 4$. Moreover, if the morphism $\Phi_{\vert 4L\vert}$ is not birational onto its image and $h^0(X,L)\geq 2$, then $L^3=1$. As an application, if $Y$ is a $n$-dimensional Fano manifold such that $-K_Y=(n-3)H$ for some ample divisor $H$, then $\vert mH\vert$ is basepoint free for $m\geq 4$ and if the morphism $\Phi_{\vert 4H\vert}$ is not birational onto its image, then $Y$ is either a weighted hypersurface of degree $10$ in the weighted projective space $\mathbb{P}(1,\cdots,1,2,5)$ or $h^0(Y,H)=n-2$.