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.

Families of log canonically polarized varieties

Abstract: Determining the number of singular fibers in a family of varieties over a curve is a generalization of Shafarevich’s Conjecture and has implications for the types of subvarieties that can appear in the corresponding moduli stack. We consider families of log canonically polarized varieties over \({\mathbb {P}^1}\) , i.e. families \({g:(Y, D) \to \mathbb {P}^1}\) where D is an effective snc divisor and the sheaf \({\omega_{Y/\mathbb {P}^1}(D)}\) is g-ample. After first defining what it means for fibers of such a family to be singular, we show that with the addition of certain mild hypotheses (the fibers have finite automorphism group, \({\mathcal {O}_Y(D)}\) is semi-ample, and the components of D must avoid the singular locus of the fibers and intersect the fibers transversely), such a family must either be isotrivial or contain at least 3 singular fibers.

Multi-modulus frequency dividers

Abstract: Various embodiments relate to multi-modulus frequency dividers, devices including the same, and associated methods of operation. A method of operating a multi-modulus divider (MMD) may include determining a common state for the MMD, wherein the MMD is configured to enter the common state regardless of a divisor value applied to the MMD. The method may further include receiving an integer value at the MMD. Further, the method may include setting the divisor value equal to the integer value. The method may also include receiving an input signal at a first frequency and generating an output signal at a second, lower frequency based on the divisor value. The method may also include receiving a second integer value at the MMD. The method may further include setting the divisor value equal to the second integer value in response to a detected current state of the MMD matching the common state for the MMD.

Symplectic Structures on Moduli Spaces of Parabolic Higgs Bundles and Hilbert Scheme

Abstract: Parabolic triples of the form $(E_*,\theta,\sigma)$ are considered, where $(E_*,\theta)$ is a parabolic Higgs bundle on a given compact Riemann surface $X$ with parabolic structure on a fixed divisor $S$, and $\sigma$ is a nonzero section of the underlying vector bundle. Sending such a triple to the Higgs bundle $(E_*,\theta)$ a map from the moduli space of stable parabolic triples to the moduli space of stable parabolic Higgs bundles is obtained. The pull back, by this map, of the symplectic form on the moduli space of stable parabolic Higgs bundles will be denoted by $\text{d}\Omega'$. On the other hand, there is a map from the moduli space of stable parabolic triples to a Hilbert scheme $\text{Hilb}^\delta(Z)$, where $Z$ denotes the total space of the line bundle $K_X\otimes{\mathcal O}_X(S)$, that sends a triple $(E_*,\theta,\sigma)$ to the divisor defined by the section $\sigma$ on the spectral curve corresponding to the parabolic Higgs bundle $(E_*,\theta)$. Using this map and a meromorphic one--form on $\text{Hilb}^\delta(Z)$, a natural two--form on the moduli space of stable parabolic triples is constructed. It is shown here that this form coincides with the above mentioned form $\text{d}\Omega'$.

Estimation of the sum of values of the divisor function

Abstract: The problem of estimation of the sum of values of a divisor function is considered in the paper. The previously known estimate is improved and the result is generalized to the case of divisor function values raised into a given power.