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.

Ultrasonic flow meter and method

Abstract: An ultrasonic flow meter (10) has a pair of transducers (36, 38). A first (36) of the pair of transducers is coupled to a first divider (18) having a first predetermined divisor. A second (38) of the pair of transducers is coupled to a second divider (24) havig a second predetermined divisor. The second predetermined divisor is not equal to the first predetermined divisor. An input (16) of the first divider (18) is coupled to a first oscillator (12). An input (22) of the second divider (24) is coupled to a second oscillator (14). A decoder circuit (50) is coupled to an output (15) of the first oscillator (12) and an output (20) of the second oscillator (14). The decoder circuit (50) determines a difference frequency.

Infinite bubbling in non-K\"ahlerian geometry

Abstract: In a holomorphic family $(X_b)_{b\in B}$ of non-K\"ahlerian compact manifolds, the holomorphic curves representing a fixed 2-homology class do not form a proper family in general. The deep source of this fundamental difficulty in non-K\"ahler geometry is the {\it explosion of the area} phenomenon: the area of a curve $C_b\subset X_b$ in a fixed 2-homology class can diverge as $b\to b_0$. This phenomenon occurs frequently in the deformation theory of class VII surfaces. For instance it is well known that any minimal GSS surface $X_0$ is a degeneration of a 1-parameter family of simply blown up primary Hopf surfaces $(X_z)_{z\in D\setminus\{0\}}$, so one obtains non-proper families of exceptional divisors $E_z\subset X_z$ whose area diverge as $z\to 0$. Our main goal is to study in detail this non-properness phenomenon in the case of class VII surfaces. We will prove that, under certain technical assumptions, a lift $\widetilde E_z$ of $E_z$ in the universal cover $\widetilde X_z$ does converge to an effective divisor $\widetilde E_0$ in $\widetilde X_0$, but this limit divisor is not compact. We prove that this limit divisor is always bounded towards the pseudo-convex end of $\widetilde X_0$ and that, when $X_0$ is a a minimal surface with global spherical shell, it is given by an infinite series of {\it compact} rational curves, whose coefficients can be computed explicitly. This phenomenon - degeneration of a family of compact curves to an infinite union of compact curves - should be called {\it infinite bubbling}. We believe that such a decomposition result holds for any family of class VII surfaces whose generic fiber is a blown up primary Hopf surface. This statement would have important consequences for the classification of class VII surfaces.

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.

Irreducible cone spherical metrics and stable extensions of two line bundles

Abstract: A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in ${\rm U(1)}$. By using the theory of indigenous bundles, we construct on a compact Riemann surface $X$ of genus $g_X \geq 1$ a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in $2 \pi \mathbb{Z}_{>1}$, which is generically injective in the algebro-geometric sense as $g_X \geq 2$. As an application, we prove the following two results about irreducible metrics: $\bullet$ as $g_X \geq 2$ and $d$ is even and greater than $12g_X - 7$, the effective divisors of degree $d$ which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension $\geq 2(d+3-3g_X)$ in ${\rm Sym}^d(X)$; $\bullet$ as $g_X \geq 1$, for almost every effective divisor $D$ of degree odd and greater than $2g_X-2$ on $X$, there exist finitely many cone spherical metrics representing $D$.