Questions tagged [normalization]

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Embedded normalization

Let $S$ be an irreducible surface in a 3-dimensional variety $X$ (everything taking place over $\mathbb{C}$, say). By Hironaka's therorem, we know for sure that there is an embedded resolution of $S$, ...
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0answers
65 views

Normalized Laplacian matrix versus walk Laplacian matrix (or normalized adjacency matrix versus walk adjacency matrix)

In graphs, found that two different normalization matrices exist for Laplacian and adiacency matrix. I will ask about the adjacency matrix (for the Laplacian matrix the questions are the same). The ...
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2answers
269 views

Why does a complex linear normalization of a real algebraic surface inherit a real structure?

Could you recommend any references to (some of) the following very basic assertions in algebraic geometry? (It seems unreasonable to reprove them in a research paper.) (1) Let a surface $X$ in $\...
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0answers
76 views

Normalization of affine curves in singular surfaces

Let $X$ be a normal, isolated surface singularity with $x_0 \in X$ the unique singularity. Let $C \subset X$ be a hyperplane section i.e., defined by a single equation. Denote by $n:\widetilde{C} \to ...
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54 views

Bijective restriction of the normalization morphism

Let $X$ be an integral separated scheme of finite type over $\mathbb{C}$. Consider the normalization morphism $f:X'\rightarrow X$. Can we always find an affine open $U\subset X'$ such that $f|_U:U\...
6
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1answer
231 views

Does ampleness descend along finite maps?

First, let me emphasize that for $X$ a not-necessarily proper variety, we say that a line bundle $L$ on $X$ is ample, if for some positive integer $n$, $L^{\otimes n}$ arises as $j^*O(1)$ for some (...
12
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1answer
278 views

Is height preserved in a normalization?

Let $R$ be a domain and $\tilde R$ its integral closure in its fraction field: $R\subset \tilde R\subset Frac(R)$. Is it true that a prime ideal $ \tilde {\mathfrak p} \subset \tilde R$ and its ...
3
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1answer
154 views

$\widetilde{R}=\bigcap_{\mathsf{ht}(\mathfrak p)=1}R_\mathfrak p$

As we know every normal Noetherian domain $R$ can be written as $$R=\bigcap_{\mathsf{ht}(\mathfrak p)=1}R_\mathfrak p.$$ I'm asking myself the following question: Question: If the normalization of $\...
4
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1answer
211 views

Ring of sections and normalization

Let $D$ be a base-point-free divisor on a normal projective variety $X$, and let $Y$ be the image of the morphism $f_{D}:X\rightarrow Y$ induced by $D$. Assume that $f_D$ is birational. Now, let $X(D)...
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0answers
341 views

Picard group of normalization

Let $X$ be a projective variety with at worst (analytic) normal crossings singularities and $\pi:\tilde{X} \to X$ be the normalisation. Is there a "nice" description relating the picard group of $X$ ...
4
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0answers
110 views

Positivity of the ramification divisor

Let $X$ be a non-normal surface such that $K_X$ is a pseudo-effective divisor and ${\rm Bs}_{-}(K_X)$ (the diminished base locus of $K_X$) equals, at least set-theoretically, the non-normal locus of $...
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3answers
448 views

How to handle sums in Tait's reducibility proof of strong normalisation?

I've been reading Girard et al's 'Proofs and Types', which in Chapter 6 presents a proof of strong normalisation for the simply typed lambda calculus with products and base types. The proof is based ...
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0answers
43 views

A question about the prediction error

I am reading about the prediction error estimation and I found the following: Suppose we have ${\mathbf{Y}}=\mathbf{x}_0+ \epsilon$, where, $\epsilon$ is normally distributed as $\sim \mathcal{N}(0, \...
3
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0answers
204 views

Understanding Strong Normalization for Identity Types in Martin-Löf Intensional Type Theory [closed]

Roughly, the strong normalization property for Martin-Löf Intensional Type Theory (MITT) tells us that every closed term $t$ of type $M$ will eventually reach a canonical normal form $t’$ such that it ...
2
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1answer
185 views

If $X$ has non-singular normalization $\dim (\mathrm{Sing(X)})=\dim (X)-1$?

Let $X\subseteq\mathbb{P}^{N}$ be a quasiprojective variety of dimension $N-1$, and let $$ \nu:X^{\nu}\rightarrow X $$ be its normalization. Let us suppose that $X^{\nu}(\neq X)$ is smooth. I wonder ...
3
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0answers
268 views

the normalized blowup

Let $X$ be a normal variety over $\mathbb{C}$ and $x\in X$ a singular point. Let $f:Y^{\nu}\to X$ be the normalized blowup at $x\in X$. (i.e. $f$ is a composition of the blowup $Y:=Bl_xX\to X$ and ...
2
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2answers
242 views

Normalization of a Noetherian local domain and line bundles on the punctured spectrum

Let $A$ be a Noetherian local domain ($2$-dimensional if needed) such that its punctured spectrum $U$ is regular, and let $A'$ be the normalization of $A$. 1) Is it possible for $A'$ to have ...
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0answers
196 views

Twisting locally free sheaves in characteristic $p$

Let $X$ be an irreducible nodal projective curve over an algebraically closed field of characteristic $p>0$. Denote by $\pi:\tilde{X} \to X$ the normalization of $X$. Recall, the short exact ...
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1answer
230 views

Projective normality of cones over projectively normal varieties

Let $X\subseteq\mathbb{P}^n$ be a smooth subvariety, with homogeneus ideal $I\subseteq k[x_0,\ldots,x_n]$. Let $C(X)\subseteq\mathbb{P}^{n+1}$ be the projective cone over $X$, so that $C(X)$ is ...
3
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1answer
198 views

Can height one maximal ideals in the normalization contract to non-height one primes in the base?

Let $R$ be a local (Noetherian) integral domain of dimension greater than one. Can the integral closure (i.e. normalization) of $R$ have a maximal ideal of height one?
3
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1answer
146 views

Eigenfunctions to 2nd-order Differential Operators: Relation between Frobenius Series Solution and Eigenfunction Normalised to the Delta Function

Consider the 2nd-order linear ODE $x f^{''}(x) + x (\beta - 2 \alpha x) \kappa / \sigma f^{'}(x) - 1 / \sigma \left[ 2 \alpha \kappa - \lambda^2 (\beta - 2 \alpha x)^2 \right] f(x) = 0$, where $\sigma&...
5
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1answer
385 views

Normalization of a curve and push forward of vector bundles

Let $C$ be a projective curve (over an algebraically closed field, not necessarily of characteristic zero) which is smooth except for exact one node. Let $\pi:\tilde{C} \to C$ be its normalization. ...
8
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2answers
552 views

Equivariant normalization?

Let $G=\mathrm{Gl}_n\mathbb C$ and let $X$ be an affine $G$-variety. Let $\phi:\tilde X\to X$ be the normalization of $X$, i.e. the spectrum of the integral closure of $\mathbb C[X]$ in its fraction ...
4
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1answer
480 views

When is normalization functorial?

Let $X$ and $Y$ be two irreducible, affine $\newcommand{\C}{\mathbb C}\C$-varieties. Let $f:X\to Y$ be a morphism. Denote by $u:\tilde X\to X$ and $v:\tilde Y\to Y$ their normalizations. Now, if $f$ ...
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2answers
2k views

meaning of normalization

I have seen the following construction and I would be very happy if someone could explain its meaning to me. We start from a smooth projective algebraic variety $X$ over a field of characteristic ...
6
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1answer
1k views

Relation between blowup and normalization

Let $X$ be a variety over an algebraically closed field with null characteristic. Let $C$ be a smooth subvariety of $X$ of dimension 1, and let $x$ be a point of $C$. We assume that $X$ is ...
6
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3answers
1k views

computational complexity of primitive recursive functions

If we have a rewrite system for primitive recursive functions, which simplifies each term according to how the function was defined, then what is the computational complexity of this calculation? That ...
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0answers
267 views

L_2-norm representation

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$. I am wondering if one can get nice representation of $L^2$-...
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1answer
652 views

Is this function field extension a Galois extension ?

Setting and question Let $X$ be a variety over an algebraically closed field of null characteristic, and let $C$ be a (regular if you want) curve included in $X$. Consider $X'$ the normalization of $...
6
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1answer
1k views

Noether normalization vs. normalization of varieties

As far as I can tell, Noether normalization uses the term "normalization" in the English sense, that something has been given a standard form. And as such it's not intrinsically related to ...
4
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1answer
517 views

Finiteness of normalization of Noetherian normal domain

I have the following question: Let $A$ be an integrally closed Noetherian domain, $K$ its field of fractions. let $L$ be a finite extension of $K$, and $B$ the integral closure of $A$ inside $L$. Is ...
3
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1answer
446 views

Which monomial subalgebras are direct summands of polynomial rings

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring, and $A:=k[x^{u^{(1)}}, \dots x^{u^{(l)}}]$ a monomial subalgebra, generated by monomials $x^{u^{(i)}} = \prod_{j=1}^n x_j^{u^{(i)}_{j}}$ with $u^{(i)} \...
3
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1answer
183 views

Simple reference for valuative criterion of integrality?

I'd like to see a complete proof of the simplest version of the following rough statement: "If $f/g$ is a rational function on a reduced scheme ($g$ not a zero divisor), and $f/g$ doesn't have poles ...
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0answers
231 views

Does the normalization of a projective morphism determine the line bundle?

Let $X$ be a smooth, complete algebraic variety and suppose I have two projective, birational morphisms $$f:X \to \mathbb{P}^n$$ and $$g:X \to \mathbb{P}^m,$$ such that the image of $f$ is the ...
2
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1answer
825 views

Line bundles, linear systems and normalization

One example that I always have in mind is that the plane nodal (or even the plane cuspidal) cubic curve $X$ is obtained by an appropirate 2-dim linear subsystem of $|\mathcal{O} (3)|$ on $\mathbb{P}^...
2
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1answer
284 views

On the normalization and the quotient of the structure sheaves

Let $\nu:\tilde{X}\to X$ be the normalization of a projective variety with non-isolated singularity. The usual object to consider is $\nu_*\mathcal{O}_{\tilde{X}}/\mathcal{O}_X$. For example, one ...
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4answers
2k views

Flatness of normalization

Let $X$ be a noetherian integral scheme and let $f \colon X' \to X$ be the normalization morphism. It is known that, if non trivial, $f$ is never flat (see Liu, example 4.3.5). What happens if we ...
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2answers
506 views

Dimensionality of a map — distance

Hello, I am looking for some words to describe what going on here. I'm sure this is not an original thought, so I'd like to read up on more from others who have thought out this topic further. FORMAT ...
7
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1answer
561 views

Normality of a locus of points in projective space

Let $U_{d,n}\subseteq(\mathbb{P}^d)^n$ denote the locus of $n$-distinct points in projective space $\mathbb{P}^d$ that lie on a rational normal curve of degree $d$, and let $V_{d,n}$ denote its ...
8
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2answers
381 views

Doing explicit computations with coordinate rings

Suppose that we are given an integral $k$-algebra $A$ of finite type explicitly, by which I mean that we are given the generators of the defining ideal $J$ where $A = k[x_1,...,x_n]/J$. What kinds of ...
3
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2answers
474 views

Is weak normality stable under completion?

I'm curious if anyone knows a reference for the following. It seems like someone must have done this somewhere, but I couldn't find a reference. Recall that an excellent reduced noetherian ring $R$ ...
8
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5answers
54k views

How do I convert a uniform value in [0,1) to a standard normal (Gaussian) distribution value?

I have uniform value in [0,1). I'd like to transform it into a standard normal distribution value, in a deterministic fashion. What I'm confused about with the Box-Muller transform is that it takes ...
4
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2answers
506 views

Normality of an affine semigroup

An affine monoid is a finitely generated commutative submonoid of $\mathbb Z^k$ for some positive integer k. Let S be an affine monoid and let G(S) be the group generated by S. We say the monoid S is ...
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6answers
6k views

What is the universal property of normalization?

What is the universal property of normalization? I'm looking for an answer something like If X is a scheme and Y→X is its normalization, then the morphism Y→X has property P and any ...