Questions tagged [normalization]

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Normal form of boolean expressions linear/affine w.r.t. conjunction and disjunction

$\DeclareMathOperator\Bool{Bool}$I am interested in boolean expressions that are linear/affine in the following sense. Let $\Bool(X)$ be the free boolean algebra over the set $X$. We can consider the ...
anuyts's user avatar
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3 votes
0 answers
99 views

When does weak normalization imply strong normalization?

Is there a possibility to get strong normalization for some kind of $\lambda$-calculus out of weak normalization with some other assumptions? For example: The term $(\lambda_y z)((\lambda_x xx)(\...
Zermelo-Fraenkel's user avatar
2 votes
1 answer
187 views

When is the singularity of a semi-normal variety a double point singularity

Let $X$ be a semi-normal projective variety and $p: \widetilde{X} \to X$ be the normalization. Suppose that $\widetilde{X}$ is smooth and there exists two smooth divisors $D_1, D_2 \subset \widetilde{...
Chen's user avatar
  • 1,583
4 votes
1 answer
97 views

Normal form of framed links under Kirby moves

It's well known that any oriented closed 3-manifold (topological or smooth) can be obtained by surgerizing along a (framed oriented) link $L$ in the 3-sphere $S^{3}$. Even better, Kirby found a ...
Student's user avatar
  • 5,008
7 votes
1 answer
328 views

Criterion for the consistency of pure type systems

Pure type systems are characterized in an almost combinatorial way: a set of axioms $\star_i : \star_j$, and a set of triples $(\star_i, \star_j, \star_k)$ saying when the dependent product $\prod_{x :...
Trebor's user avatar
  • 971
1 vote
0 answers
181 views

Moduli interpretation of normalization of moduli space

The question is about formal and rigid geometry, but I would be interested in an answer from an algebraic geometry point of view as well. Let $\mathfrak{X}$ be a formal moduli space (e.g., the formal ...
Jon Aycock's user avatar
3 votes
0 answers
254 views

If the normalization is affine, is it affine? (if quasiaffine)

I was surprised to find out that, even if the normalization $X^\nu$ of a scheme $X$ is affine, $X$ may not be affine (remove the line $x=y$ from their example to make the source affine). In the ...
Leo Herr's user avatar
  • 1,084
8 votes
0 answers
317 views

Example of torsion differential forms

I am looking for an example of a normal affine variety $V$ over a perfect field $k$ such that the differentials $\Omega_{V/k}$ are not torsion free. If normality is not required, an example is given ...
Katharina Hübner's user avatar
4 votes
0 answers
111 views

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 $S$, ...
C. Gachet's user avatar
2 votes
0 answers
422 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 ...
volperossa's user avatar
2 votes
2 answers
343 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 $\...
Mikhail Skopenkov's user avatar
2 votes
0 answers
151 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 ...
Jana's user avatar
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0 votes
0 answers
78 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\...
mikhalych's user avatar
6 votes
1 answer
262 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 (...
jacob's user avatar
  • 2,814
12 votes
1 answer
388 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 ...
Georges Elencwajg's user avatar
3 votes
1 answer
175 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 $\...
Vincenzo Zaccaro's user avatar
3 votes
1 answer
277 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)...
user avatar
5 votes
0 answers
615 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 $X$ ...
Ron's user avatar
  • 2,116
4 votes
0 answers
135 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 $...
Joaquín Moraga's user avatar
8 votes
3 answers
960 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 ...
RAC's user avatar
  • 83
1 vote
0 answers
48 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, \...
neda's user avatar
  • 11
3 votes
0 answers
256 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 ...
StudentType's user avatar
2 votes
1 answer
223 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 ...
JosuaJones's user avatar
3 votes
0 answers
482 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 ...
Beankien's user avatar
2 votes
2 answers
314 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 ...
O-Ren Ishii's user avatar
1 vote
0 answers
203 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 ...
user43198's user avatar
  • 1,949
1 vote
1 answer
401 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 ...
gio's user avatar
  • 1,149
3 votes
1 answer
685 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?
Neil Epstein's user avatar
  • 1,752
3 votes
1 answer
204 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&...
user51524's user avatar
5 votes
1 answer
650 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. ...
user46578's user avatar
  • 823
10 votes
2 answers
746 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 ...
Jesko Hüttenhain's user avatar
4 votes
1 answer
634 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 $f$ ...
Jesko Hüttenhain's user avatar
2 votes
2 answers
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 ...
normali's user avatar
  • 23
7 votes
1 answer
2k 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 ...
Lierre's user avatar
  • 1,044
6 votes
3 answers
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 ...
AKS's user avatar
  • 63
0 votes
0 answers
271 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 $L^2$-...
David's user avatar
  • 71
1 vote
1 answer
834 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 $...
Lierre's user avatar
  • 1,044
6 votes
1 answer
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 ...
Allen Knutson's user avatar
4 votes
1 answer
706 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 ...
Sasha's user avatar
  • 5,492
3 votes
1 answer
536 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)} \...
Thomas Kahle's user avatar
  • 1,961
3 votes
1 answer
215 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 ...
Allen Knutson's user avatar
0 votes
0 answers
247 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 ...
Zaky's user avatar
  • 1
2 votes
1 answer
1k 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}^...
Jodel's user avatar
  • 23
2 votes
1 answer
337 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 ...
Dmitry Kerner's user avatar
18 votes
4 answers
4k 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 ...
Ricky's user avatar
  • 3,674
1 vote
2 answers
528 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 ...
William Entriken's user avatar
7 votes
1 answer
604 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 ...
Noah Giansiracusa's user avatar
8 votes
2 answers
414 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 ...
user332's user avatar
  • 3,878
3 votes
2 answers
533 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 $R$ ...
Karl Schwede's user avatar
  • 20.2k
13 votes
5 answers
69k 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 ...
Joseph Turian's user avatar