## Localization of multiplicity in algebraic geometry

first a disclaimer: I am not an expert in alg. geometry so please don’t shoot. Suppose X is a closed subscheme (not nec. reduced, and dim>0) of a smooth (projective if you want) variety Y. Suppose (for simplicity) that X is pure dimensional and Z is an irreducible component of X. The multiplicity of Y … Read more

## Is there a classification of higher-degree generalisations of confocal conic sections?

The 1-parameter families of ellipses and hyperbolas with a given pair of points in the plane as their foci yield “orthogonal double-foliations” of the plane. That is, once the foci are specified, any point in the plane lies on both a unique ellipse and a unique hyperbola, which are orthogonal at the point of intersection. … Read more

## Is the Serre dualizing complex local in the analytic topology?

There is a Serre dualizing complex $S_X\in D^b Coh(X)$ for any scheme $X$ of finite type. For proper schemes, it is characterized as the sheaf defining the right adjoint to derived global sections, and for a general scheme $U$, one can define $S_U: = j^*S_X$ for $j:U\to X$ the open embedding into some choice of … Read more

## GCD in polynomial vs. formal power series rings

I’m having problems finding an appropriate reference for this question. Given two elements $f, g \in \mathbb{C}[x_1, \dots, x_n]$, consider their greatest common divisor, $\gcd_{\mathbb{C}[x_1, \dots, x_n]}(f, g) = h \in \mathbb{C}[x_1, \dots, x_n]$. Now, consider this two elements $f$ and $g$ as elements of $\mathbb{C}[[x_1, \dots x_n]]$, i.e., formal power series with finitely many … Read more

## Simple maps: Flat versus locally trivial

In deformation of complex analytic spaces, one usually considers an analytic proper simple surjective map \varpi: \mathscr{M} \twoheadrightarrow \mathscr{P} as an analytic family. However the term simple differs by literature. There are two notions of simple: as a locally trivial map and as a flat map. Are these definitions equivalent? I should mention too that … Read more

## English reference for Douady/Grauert construction of versal deformations of compact complex spaces

I’m trying to learn about the deformation theory of compact complex spaces. I’m familiar with the case of compact complex manifolds from the paper “On the Locally Complete Families of Complex Analytic Structures” by M. Kuranishi. The original references for the case of compact complex spaces seem to be the papers “Le problème des modules … Read more

## Can an analytic variety extend along a codimension 2 subvariety?

Let X be a smooth, connected, complex analytic variety, and Y⊂X a closed, analytic subvariety of codimension at least 2. Now let V⊂X∖Y be a closed, analytic subvariety. Is the closure ˉV of V in X also analytic? Motivation: In the case where X is a surface, this seems to follow from Riemanns extension theorem, … Read more

## Lattice points close to a line

Take a sheet of grid paper and draw a straight line in any direction from the origin. What is the closest non-zero grid point \boldsymbol{p}\in\mathbb{Z}^2 within a distance \epsilon>0 of the line? I have been unable to construct an example where the distance is large for any \epsilon. I would like to prove the following: … Read more

## supporting facts to fujita conjecture

I came across the Fujita conjecture which is perhaps very widely known. I want to know what are the supporting facts to the truth of the conjecture. http://en.wikipedia.org/wiki/Fujita_conjecture Answer There are a number of things known. (1). As Libli answered, if L is globally generated and ample (for example, very ample) then KX⊗LdimX+1 is globally … Read more

## Subadditivity of multiplier ideals with a pluriharmonic function

I would like to have a reference for the following two facts (if true): Let $D$ be a nef and big divisor on an algebraic variety $X$ and $h$ a Hermitian metric with minimal singularities on $D$, write $h=e^{-\phi}$ around some point $x\in X$ with $\phi$ being the local weight. Is it true that $\phi$ … Read more