2
votes
2answers
78 views

Geometrically reduced variety

What is a geometrically reduced variety (or geometrically reduced algebraic set if you will, a variety has not been assumed to be irreducible in this definition)? I tried looking up on the internet as ...
3
votes
2answers
78 views

Closed immersion definition

Hartshorne defines a closed immersion as a morphism $f:Y\longrightarrow X$ of schemes such that a) $f$ induces a homeomorphism of $sp(Y)$ onto a closed subset of $sp(X)$, and furthermore b) the ...
4
votes
2answers
105 views

What is the zero subscheme of a section

Let $X$ be a scheme, $\mathcal F$ a locally free sheaf of rank $r$ and $s \in \Gamma(X, \mathcal F)$ a global section of $\mathcal F$. Question: What is the zero subscheme of $s$? I can't believe ...
0
votes
0answers
15 views

Is there a class of schemes s.t. morphisms from a quasi-compact (-separated) scheme $X$ to any one of them is quasi-compact (-separated)?

The question is intend for both quasi-compact or quasi-separated notions, but let me elaborate on the details of quasi-compact only. Def: A scheme $X$ is said to be quasi-compact if any of its covers ...
2
votes
1answer
33 views

Why do we want to define a $k$-scheme to be birational if the rational map (and its inverse) to $\Bbb A_k^n$ is over $k$?

Two varieties $X,Y$ are said to be birational if there exist rational maps in each direction such that either composition is the identity on a open dense subset. Note that here the morphisms aren't ...
3
votes
0answers
54 views

Zariski cotangent space, as defined in Arapura's “Algebraic Geometry over the Complex Numbers”

In "Algebraic Geometry over the Complex Numbers", Arapura gives the following definition: Definition 2.5.8. When $(R, m, k)$ is a local ring satisfying the tangent space conditions, we define its ...
1
vote
1answer
43 views

Help in this definition of morphism

I need help in this definition of morphism of affine algebraic sets which I found in a book: Let $X$ and $Y$ affine algebraic sets and say $$f:X\to X'\ \text{and}\ g:Y\to Y'$$ isomorphisms with ...
3
votes
0answers
69 views

Definition of differential forms

I am reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. The authors introduce some definitions about differential forms at the beginning of Section 2.2. These ...
2
votes
1answer
48 views

Ring of rational functions for reducible variety

Let $X$ be some affine algebraic variety over $\mathbb{k}$ (i.e. some closed subset in $\mathbb{A}_\mathbb{k}^n$). First suppose $X$ to be irreducible. Then the algebra $\mathbb{k}[X]$ is a domain and ...
1
vote
0answers
46 views

evaluating a meromorphic section of a line bundle at a point

Let $D$ be a Cartier divisor of a variety $X/K$ with associated line bundle $\mathcal{O}(D)$ and meromorphic section $s_D$. How do you define $s_D(P) \in K$ for $P \in X(K) \setminus ...
7
votes
1answer
133 views

Is taking inverse automatically well-defined?

By the usual definition, Lie group is a manifold $G$ with a group structure on it such that the multiplication $m\colon G\times G\to G$ and taking inverse $i\colon G\to G$ are both smooth maps. But it ...
5
votes
2answers
133 views

Why is a section of a sheaf over closed set defined this way?

Why is a section of a sheaf $F$ over closed set $S \subset X$ is defined as inductive limit $$ \varinjlim_{S\subset U} F(U)\; ?$$ From my point of view, we should define it as a function, which each ...
3
votes
2answers
68 views

Hartshorne Lemma(II 6.1)

Hi I am wondering what Hartshorne means by "an open affine subset on which $f$ is regular". This is part of the first sentence in the proof of lemma 6.1 in chapter two of the book "Algebraic Geometry ...
2
votes
1answer
328 views

Projective Normality

What is the significance of studying projective normality of a variety ? How does it relate to non-singularity, rationality of a variety ?
2
votes
1answer
121 views

On the definition of divisors in Riemann Surfaces

The sum notation for a Divisor $D$ in a Riemann Surface $X$ (as in Miranda's "Algebraic Curves and Riemann Surfaces") is $$ D=\sum_{p\in X} D(p)\cdot p $$ That is, $D$ assumes the value $D(p)$ at $p$. ...
5
votes
2answers
129 views

Definition of Unipotent in Positive Characteristic

Let $G$ be an affine algebraic group over an algebraically closed field $k$ whose characteristic is $p>0$. Can $\mathcal{U}(G)$, the set of unipotent elements of $G$, be characterized as all ...
3
votes
3answers
509 views

Definition of degree of finite morphism plus context

Let $f: X \rightarrow Y$ be a finite morphism of schemes, defined here, http://en.wikipedia.org/wiki/Finite_morphism I always assumed that the degree of $f$ was the degree of the induced field ...
2
votes
1answer
313 views

Definition of the Ideal Sheaf

Let $Y$ be a closed subscheme of a scheme $X$ and let $i:Y \rightarrow X$ be the inclusion morphism. Then the ideal sheaf of $Y$ is defined to be the kernel of the morphism of sheafs $i^{\#}: ...
2
votes
2answers
102 views

Separated and Finite Type Scheme over an Algebraically Closed Field

Let $(X,\mathcal{O}_X)$ be a separated scheme and of finite type over an algebraically closed field $k$. The fact that $X$ is separated means that the image of $X$ under the diagonal morphism $\Delta: ...
1
vote
1answer
150 views

Definition of tautological section.

Reading Barth, Peters: Compact complex surfaces, i stumbled across the following: Let $Y$ be an algebraic surface over $k =\mathbb{C}$, and $\mathcal{L}$ an invertible sheaf on $Y$. Denote by $p: L ...
3
votes
1answer
299 views

“sheaf” au sens de Serre

I learned the definition of sheaves from Algebraic Geometry by Hartshorne, while reading Serre's GAGA, I was wondering if there was another definition of sheaves. [Here is the link of the English ...
4
votes
1answer
160 views

intrinsic and geometric definition of blow-up

Suppose I am given an algebraic variety $X$ and a (closed) point $x \in X$. I know of two descriptions of the blow-up of $X$ at $x$. One is intrinsic but not geometric: if $\mathcal{I}_x$ denotes the ...
3
votes
0answers
78 views

Closed / embedded surface

Given a closed surface in $\mathbb R^3$, is it necessarily an "embedded surface"? I think it is true, but that is just because I can't think of a closed surface for which we cannot construct a smooth ...
6
votes
1answer
538 views

Why does the definition of an open subscheme / open immersion of schemes allow for an “extra” isomorphism?

After taking an algebraic geometry course last year, I've been reviewing the material this year, and I remembered something that struck me as odd, but which I'd neglected to ask about at the time: ...
7
votes
3answers
665 views

What is the operation $\boxtimes$?

Reading papers about $p$-adic analysis and Galois representations, I have found objects like this $D \boxtimes \mathbb{Q}_p$. So my question is what is $\boxtimes$ and how do we read it ?