Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [algebraic-geometry]

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

29074 questions
5
votes
0 answers

What is an algebraic scheme?

In Sernesi's book "Deformations of Algebraic Schemes", on page 2 ("Terminology and Notation"), he defines "(schemes)" to be the category of locally noetherian separated $k$-schemes, where $k$ is a fixed alg. closed field. Then he writes…
oxeimon
  • 12,279
5
votes
1 answer

Relation of Function Field of a scheme to the Local Ring of its Prime Divisor

Refer to p. 130 in Hartshorne: Let $X$ be a noetherian, integral separated scheme, regular in codimension 1, and let $Y$ be a prime divisor of $X$, with generic point $\eta$. Let $\xi$ be the generic point of $X$ and $K=\mathcal{O}_{X,\xi}$ is the…
Manos
  • 25,833
5
votes
1 answer

Dimension of affine scheme

I'm reading Eisenbud's "Geometry of Schemes" now. In the book, dimension of a scheme $X$ is defined by supremum of local dimension $\dim(X, x)$, where $\dim(X, x)$ is Krull dimension of stalk $\mathcal{O}_{X, x}$. Also, $x$ is "singular " if local…
Seewoo Lee
  • 15,137
  • 2
  • 19
  • 49
5
votes
1 answer

regular functions definition

In the literature there appear to be two different definitions of "regular functions": defined locally by polynomials https://en.wikipedia.org/wiki/Morphism_of_algebraic_varieties defined locally by rational functions as a well defined quotient of…
user352102
  • 904
5
votes
1 answer

Canonical divisor of the projective line

I want to calculate a cannonical divisor of $\mathbb{P}^1_k$. We have the regular function $f=id:\mathbb{P}^1\rightarrow\mathbb{P}^1$. Thus we get a regular differential form $df=f-f(x)\text{ mod }m_p^2.$ But how can we compute $div f?$ What is…
user404105
5
votes
1 answer

Moduli of Riemann surfaces (genus g curves) is a variety.

I often see the moduli spaces $\mathcal{M}_g$, or at least the coarse moduli space, of Riemann surfaces of genus $g$ described as the set of isomorphism classes of Riemann surfaces of genus $g$. Obviously, the moduli space has more structure than…
user7090
  • 5,453
  • 1
  • 22
  • 53
5
votes
0 answers

A question about radical ideal

Given an ideal $I=\langle f_1,\ldots,f_s\rangle\subseteq\mathbb{C}[X_1,\ldots,X_n]$, suppose that the differentials of its generators $df_1,\ldots,df_s$ are linearly independent at any point $x\in\mathbb{V}_\mathbb{C}(I)$, the algebraic set of $I$…
jeff
  • 59
5
votes
2 answers

Finite morphism of varieties - morphism of sheaves

Let $f:X\rightarrow Y$ be a finite morphism of non-singular projective varieties of degree $d$. Consider the map of sheaves $O_Y\rightarrow f_*O_X$. Is this morphism injective? Why? $f_*O_X$ is a rank $d$ vector bundle. Suppose the above morphism…
user52991
  • 708
5
votes
3 answers

Affine space over non algebraically closed field

Re-edited: Let $k$ be a field, not necessarily algebraically closed. Then what is the relation of the affine space $\mathbb{A}^n(k)$ with $k^n$ or $\bar{k}^n$? Note: I am quite confused about what an affine space is, so i am asking this question,…
Manos
  • 25,833
5
votes
1 answer

When the the presheaf of image of morphism of sheaves is a sheaf?

For given a morphism of sheaves, in general, I know that the presheaf of image(or the presheaf of cokernel) is not a sheaf. is there when the the presheaf of image(or the presheaf of cokernel) of morphism of sheaves is a sheaf?
Sang Cheol Lee
  • 3,119
5
votes
0 answers

Definition of closed immersion.

Here is the definition of closed immersion given on Stacks Project. In Hartshorne (II, Section 3), a closed immersion of schemes is only defined by the first two properties. What does "locally generated by sections" mean? And why is it considered…
D_S
  • 33,891
5
votes
2 answers

Why is the residue field of a $k$-scheme an extension of $k$?

Let $k$ be any field and let $X$ be a scheme over $k$. For each $x$, the residue field $\kappa(x)$ is a field extension of $k$. Why is this true? My understanding is that a scheme over a field really just amounts to a ring homomorphism $k…
Not a grad student
  • 1,638
5
votes
0 answers

Zero subscheme of a section: Making computations.

In the Algebraic Geometry course I am following we have defined vector bundles in the following way: Given a locally free sheaf $\mathscr{E}$ on a scheme $S$ we can define a functor, $$\mathbb{V}(\mathscr{E}): \operatorname{Sch}/S \to…
Abellan
  • 3,723
5
votes
2 answers

Examples of quasi-projective varieties that are not (topologically) quasi-affine

I'm trying to think of a quasi-projective variety that is not isomorphic to a quasi-affine one. I image that it must be $Y \subseteq \mathbb{P}^n$ of at least $n \geq 3$, and maybe $\operatorname{dim} Y \geq 2$ as well. I am also interested in…
basket
  • 2,077
5
votes
1 answer

Why is regular function?

Suppose that we have two varieties $V,W$ (affine or projective, arbitrary). They are two algebraic object and as usual, we can define the map $\varphi: V\longrightarrow W$. It's easy to think about it as…
Arsenaler
  • 3,930
1 2 3
…
99 100