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.

Infinitesimal deformations of a Calabi-Yau hypersurface of the moduli space of stable vector bundles over a curve

Abstract: Let $X$ be a compact connected Riemann surface of genus $g$, with $g\geq 2$, and ${\cal M}_{\xi}$ a smooth moduli space of fixed determinant semistable vector bundles of rank $n$, with $n\geq 2$, over $X$. Take a smooth anticanonical divisor $D$ on ${\cal M}_{\xi}$. So $D$ is a Calabi-Yau variety. We compute the number of moduli of $D$, namely $\dim H^1(D, T_D)$, to be $3g-4 + \dim H^0({\cal M}_{\xi}, K^{-1}_{{\cal M}_{\xi}})$. Denote by $\cal N$ the moduli space of all such pairs $(X',D')$, namely $D'$ is a smooth anticanonical divisor on a smooth moduli space of semistable vector bundles over the Riemann surface $X'$. It turns out that the Kodaira-Spencer map from the tangent space to $\cal N$, at the point represented by the pair $(X,D)$, to $H^1(D, T_D)$ is an isomorphism. This is proved under the assumption that if $g =2$, then $n eq 2,3$, and if $g=3$, then $n eq 2$.

A sharper probability estimate for Shor's algorithm

Abstract: Let N be a (large positive integer, let b > 1 be an integer relatively prime to N, and let r be the order of b modulo N. Finally, let QC be a quantum computer whose input register has the size specified in Shor's original description of his order-finding algorithm. We prove that when Shor's algorithm is implemented on QC, then the probability P of obtaining a (nontrivial) divisor of r exceeds 0.7 whenever N exceeds 2^{11}-1 and r exceeds 39, and we establish that 0.7736 is an asymptotic lower bound for P. When N is not a power of an odd prime, Gerjuoy has shown that P exceeds 90 percent for N and r sufficiently large. We give easily checked conditions on N and r for this 90 percent threshold to hold, and we establish an asymptotic lower bound for P of (2/Pi) Si(4Pi), about .9499, in this situation. More generally, for any nonnegative integer q, we show that when QC(q) is a quantum computer whose input register has q more qubits than does QC, and Shor's algorithm is run on QC(q), then an asymptotic lower bound on P is (2/Pi) Si(2^(q+2) Pi) (if N is not a power of an odd prime). Our arguments are elementary and our lower bounds on P are carefully justified.

Semistability of Frobenius Direct Image of Representations of Cotangent Bundles

Abstract: Let k be an algebraically closed field of characteristic p > 0, X a smooth projective variety over k with a fixed ample divisor H, FX : X → X the absolute Frobenius morphism on X Let E be a rational GLn(k)-bundle on X, and ρ : GLn(k) → GLm(k) a rational GLn(k)-representation of degree at most d such that ρ maps the radical RGLn(k)) of GLn(k) into the radical R(GLm(k)) of GLm(k) We show that if $$F_X^{N*}(E)$$ is semistable for some integer $$N \ge {\max {_{0 < r < m}}}(_r^m) \cdot {\log _p}(dr)$$ , then the induced rational GLm(k)-bundle E(GLm(k)) is semistable As an application, if dimX = n, we get a sufficient condition for the semistability of Frobenius direct image $$F_{X*}(\rho*(\Omega_X^1))$$ , where $$\rho*(\Omega_X^1)$$ is the vector bundle obtained from $$\Omega_X^1$$ via the rational representation ρ

Maximal sets of integers with distinct divisors

Abstract: A set of positive integers is said to have the distinct divisor property if there is an injective map that sends every integer in the set to one of its proper divisors. In 1983, P. Erdős and C. Pomerance showed that for every $c>1$, a largest subset of $[N,cN]$ with the distinct divisor property has cardinality $\sim \delta(c)N$, for some constant $\delta(c)>0$. They conjectured that $\delta(c)\sim c/2$ as $c \to \infty$. We prove their conjecture. In fact we show that there exist positive absolute constants $D_1,D_2$ such that $D_1\le c^{\beta}(c/2-\delta(c))\le D_2$ where $\beta = \log 2/\log (3/2)$.

Omega results for the divisor and circle problems

Abstract: We obtain new omega results for the error terms in two classical lattice point problems. These results are likely to be the best possible.