The normalization tag has no usage guidance.

**6**

votes

**1**answer

202 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 (...

**11**

votes

**1**answer

224 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**

votes

**1**answer

142 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**

votes

**1**answer

192 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)...

**5**

votes

**0**answers

180 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**

votes

**0**answers

94 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 $...

**7**

votes

**3**answers

274 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 ...

**0**

votes

**0**answers

33 views

### Normalise radial eigenfunction - Laguerre power exponential

I am trying to normalise an eigenfunction of the form
\begin{align}
\nu(r,\theta) = r^{l-n} e^{-\frac{r^2}{2C}} L_n^{l-n}\left( \frac{r^2}{C} \right)
cos((l-n) \theta),
\end{align}
...

**1**

vote

**0**answers

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**

votes

**0**answers

177 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**

votes

**1**answer

175 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 ...

**2**

votes

**0**answers

204 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**

votes

**2**answers

205 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 ...

**1**

vote

**0**answers

164 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 ...

**0**

votes

**1**answer

190 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**

votes

**1**answer

177 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**

votes

**1**answer

136 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**

votes

**1**answer

306 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. ...

**7**

votes

**2**answers

516 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**

votes

**1**answer

441 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 $f$ ...

**2**

votes

**2**answers

1k 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**

votes

**1**answer

963 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**

votes

**3**answers

928 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 ...

**0**

votes

**0**answers

265 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 $L^2$-...

**1**

vote

**1**answer

612 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**

votes

**1**answer

863 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**

votes

**1**answer

450 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**

votes

**1**answer

423 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**

votes

**1**answer

170 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 ...

**0**

votes

**0**answers

220 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**

votes

**1**answer

748 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**

votes

**1**answer

267 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 ...

**11**

votes

**4**answers

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 ...

**0**

votes

**2**answers

496 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**

votes

**1**answer

549 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**

votes

**2**answers

374 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**

votes

**2**answers

444 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 $R$ ...

**6**

votes

**5**answers

41k 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**

votes

**2**answers

483 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 ...

**23**

votes

**6**answers

5k 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 ...