3
votes
3answers
100 views

How to imagine “tensoring with Serre's twisted sheaf”

What has an algebraic geometer in mind when (s)he sees $\otimes \mathcal{O}(1)$? I think it has something to do with an intersection of a hypersurface...? Thanks, Adrian
1
vote
0answers
26 views

What is a complete intersection?

I was reading and I encountered something that goes: We have degree $d$ polynomials in $s$ variables $F_1, ..., F_n$ with coefficients in integers. Let $X$ be the complete intersection defined by the ...
3
votes
1answer
39 views

Degree of a divisor for algebraically closed fields vs not closed ones

Suppose we have an algebraically closed field $F$ and we consider the projective space $\mathbb{P}^1$ over $F$. If we consider some divisor $D = n_P P + n_Q Q +n_s S$, then we say the degree of $D$ ...
1
vote
0answers
17 views

When is a series expansion related to its derivative by a polynomial equation?

Is there some common theory behind the following two examples? Example 1. Let $p(t) = \sum_{n \geq 0} (-1)^k t^{2k}/(2k)!$, and $x = p(t), y = p'(t)$. Then $x^2 + y^2 = 1$ identically. Example 2. ...
3
votes
1answer
87 views

Intuition on formal neighborhood in a scheme

Let $X$ be a Noetherian scheme, $x \in X$ a closed point. Denote by $\hat X$ the completion of $X$ along $x$. Now assume that two coherent modules $F, G$ on $X$ coincide over $\hat X$, i.e. $i^*F = ...
3
votes
4answers
783 views

What are the conditions for a polygon to be tessellated?

Upon one of my mathematical journey's (clicking through wikipedia), I encountered one of the most beautiful geometrical concept that I have ever encountered in my 16 and a half years on this oblate ...
0
votes
1answer
50 views

Geometric $k$-blades

What are the purposes of $k$-blades? Why is it important to have a oriented area, or an oriented volume? I'm referring to $k$-blades in such a way that $$\hat{v_1} \wedge \cdots \wedge \hat{v_k} $$ ...
7
votes
2answers
314 views

Algebraic geometry in representation theory?

I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
4
votes
0answers
89 views

Is there any equivalence between the category of schemes over $\mathbb R$ and the category real manifolds

The equivalence of the category of smooth projective curves over $\mathbb C$ and the category of compact Riemann surfaces is, I believe, well documented. For example, it is mentioned on the wiki page: ...
4
votes
1answer
294 views

Intuition behind isomorphism of algebraic varieties

Let $X \subset \mathbb A^n$, $W \subset \Bbb A^m$ be two algebraic sets. A function $\phi:X \rightarrow W$ is a morphism if there exist $m$ polynomial functions $f_1,\ldots,f_m \in K[X]$ such that for ...
13
votes
3answers
656 views

How does intuition fail for higher dimensions?

From this answer: Now, Algebraic Geometry is one of the oldest, deepest, broadest and most active subjects in Mathematics with connections to almost all other branches in either a very direct ...
10
votes
2answers
340 views

Motivating (iso)morphism of varieties

I am reading course notes on algebraic geometry, where a morphism of varieties is defined as follows ($k$ is an algebraically closed field): Let $X$ be a quasi-affine or quasi-projective ...
0
votes
3answers
120 views

How to think about one-point schemes?

As topological spaces, all of $\text{Spec}(k), \text{Spec}(k(x)), \text{Spec}(k[x]/(x^2))$ and $\text{Spec}(k(x_1,\cdots,x_n))$ are all homeomorphic, since they are all one point-spaces. However, as ...
6
votes
1answer
532 views

How do mathematicians think about high dimensional geometry?

Many ideas and algorithms come from imagining points on 2d and 3d spaces. Be it in function analysis, machine learning, pattern matching and many more. How do mathematicians think about higher ...
5
votes
0answers
125 views

Why is better to work with the spectrum of prime ideals than with the maximal one, for example in the definition of affine scheme.

When we have an algebraic variety we can identifie the points of the variety with maximal ideals of the coordinate ring. I would like to know why is more natural to define the main structure of the ...
6
votes
0answers
137 views

Why are injective $\mathscr{O}$-modules flasque?

Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
3
votes
1answer
201 views

Intuition on the definition of “rational maps”

I'm studying some representation theory on $S_n$ and $GL(V)$ and tensor spaces, and have come across a lot of material involving rational representations. I'm not really an algebraic geometer by ...
26
votes
3answers
1k views

Why the emphasis on Projective Space in Algebraic Geometry?

I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed. Why does Miranda (and from what little ...
5
votes
1answer
162 views

Geometric invariants of a scheme

Following my previous question about sheaf cohomology, I'd like to ask about its applications to algebraic geometry. I have now learned a little about homological algebra and I can see that for ...
13
votes
1answer
1k views

What is the intuition behind the concept of Tate twists?

For any field $K$ we can define the cyclotomic character $\chi: \operatorname{Gal}(K)\rightarrow GL_1(\hat{\mathbb{Z}})$. For any representation $V$ (I will view this as a module over ...
14
votes
5answers
706 views

An example of a scheme in the language of schemes

Somewhat related to this question, but almost infinitely more basic. A Confession I am, should classification prove essential, a differential geometer and a topologist by inclination and by ...
17
votes
4answers
2k views

intuitive explantions for the concepts of divisor and genus

when trying to explain AG-codes to computer scientists, the major points of contention i am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. are there any ...
12
votes
4answers
2k views

Meaning of closed points of a scheme

This is a question in Liu's book. Let $X$ be a quasi-compact scheme. Show that $X$ contains a closed point. Well I'm unable to do this question, so any help would be appreciated. This question also ...
12
votes
1answer
359 views

Stacks are just sheaves up to Isomorphism

I have heard that one can think of stacks on a site as taking sheaves but instead of the restrictions being equal, we just loosen it to isomorphic, and treat the sheaf conditions with the "obvious" ...
4
votes
3answers
610 views

genericness and the Zariski topology

What does it mean (in a mathematically rigorous way) to claim something is "generic?" How does this coincide with the Zariski topology?