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
1 answer

Every affine variety in the real affine space $A^n_R$ is the zero locus of one polynomial.

Sorry if this is a stupid question to ask. This is an exercise from Gathmann's Algebraic Geometry. Show that every affine variety in the real affine space $A^n_R$ is the zero locus of one polynomial. If some context might help, he is talking about…
KittyL
  • 16,965
5
votes
1 answer

How to show that a polynomial maps an algebraic set to an algebraic set?

Sorry if this is an ignorant question. I am studying algebraic geometry. This isn't an exercise problem. It is an assumption I can use to prove something else. I think it must be obvious, but I don't know how to prove it. My attempt: Let…
KittyL
  • 16,965
5
votes
4 answers

Show that $X$ is not an affine variety

I need some help proving that $X=\{(x,x)~|~x \in \mathbb{R}, x \neq 1\}$ is not an affine variety. I would like to proceed by supposing it is an affine variety and then finding a contradiction. So assume $X=V(f_1,...,f_s)$. Now I want to show that…
Sam
  • 1,168
5
votes
2 answers

Affine algebraic subset of $\mathbb{A}_k^4$

How do I go about proving the subset $V = \{(s^3, s^2t, st^2, t^3)\mid s, t \in k\}$ is an affine algebraic subset of $\mathbb{A}_k^4$ and find $\mathbb{I}(V) \subset k[x_0, x_1, x_2, x_3]$?
user209939
5
votes
1 answer

Image of a maximal torus via epimorphism

Let $\phi \colon G \rightarrow H$ be a surjective morphism of linear algebraic groups. Let $T \subset G$ be a maximal torus: how can I prove that $\phi(T)$ is also a maximal torus? To show that $\phi(T)$ is a torus is quite easy but I cannot find an…
N.B.
  • 2,109
5
votes
1 answer

Prove that a presheaf is a sheaf

Let $X$ be a variety. Show that if $X$ is irreducible, then the constant abelian presheaf $\mathcal{F}$ with $\mathcal{F}(U)=\mathbb{Z}$ for every nonempty open subset $U\subseteq X$ and $\mathcal{F}(\emptyset)=0$ is a sheaf. Any leads? What does…
Vinyl_cape_jawa
  • 4,794
5
votes
1 answer

Why do we have two definitions of Cartier divisor?

Why do we have several definitions of Cartier divisor? For example, I found in two books the following definitions: Let $X$ be a scheme. We denote the group $H^0(X,\mathcal K_X^*/\mathcal O_X^*)$ by $\operatorname{Div}(X)$. The elements of…
student
  • 51
5
votes
1 answer

Two scheme morphisms agree on a dense open subset must be equal

I'm trying to prove this statement Let $ f,g:X\rightarrow Y$ be two $ S $-scheme morphisms that agree on $ U $, a dense open subset of $ X $. If $ X $ is reduced and $ Y$ separated, then $ f = g $. I've gone so far as showing that the locus of…
PeterM
  • 5,367
5
votes
1 answer

Etale morphism and reduced schemes

Let $f:X \to Y$ be an etale morphism of Noetherian schemes. Is it true that the induced morphism on the reduced schemes, i.e., $f_{\mathrm{red}}:X_{\mathrm{red}} \to Y_{\mathrm{red}}$ is etale as well?
user54369
  • 447
  • 2
  • 8
5
votes
2 answers

Show that if $f \in \mathbb{C} \left[x_1, \dots, x_n \right]$ vanishes at every point of $\mathbb{Z}^n$, then $f$ is the zero polynomial.

I am working on a problem from Ideas, Varieties, and Algorithms: Show that if $f \in \mathbb{C} \left[x_1, \dots, x_n \right]$ vanishes at every point of $\mathbb{Z}^n$, then $f$ is the zero polynomial. My attempt so far: This proof is by…
MathMajor
  • 6,478
5
votes
2 answers

Rational points of projective spaces over rings

Let $X=\mathbf{P}^n_A = \text{Proj} A[T_0,\ldots,T_n]$. If $A$ is a field, there is a simple classical description of $X(A)$. However, if $A$ is a more general ring, like $\mathbf{Z}$, I don't see an easy way to characterize rational points. For…
user1971
5
votes
1 answer

What is the Euler characteristic of a determinantal hypersurface?

If $X\subset \mathbb P^{n}$ is a smooth (complex) hypersurface, one can compute its topological Euler characteristic $\chi(X)$ by taking the degree of the $0$-cycle $c_{n-1}(T_X)\cap [X]$. If $X$ is singular, I do not know what can be said.…
Brenin
  • 14,072
5
votes
1 answer

smooth irreducible variety with finite (non-zero) Picard group?

Does there exist a smooth irreducible variety $X/\mathbb{C}$ such that $\mathrm{Pic}(X)$ is finite and non-zero?
adrido
  • 2,283
5
votes
4 answers

Is the projective closure of a smooth variety still smooth?

Let $X$ be a closed subscheme of $\mathbb{A^n}$ (over a basefield) defined by an ideal $I$ and consider the immersion $\mathbb{A^n}\to \mathbb{P^n}$, $(x_1,\ldots, x_n)\mapsto [x_1,\ldots,x_n,1]$. One may consider the projective variety $\bar X$ in…
Daniel Dreiberg
  • 1,309
4
votes
1 answer

When does Coh$(X)$ have enough locally frees?

Let $X$ be a scheme of finite type over $\mathbb C$. One might be interested in morphisms in the derived category $D(X)$ of coherent sheaves on $X$, that are morphisms $f:E^\bullet\to F^\bullet$ of complexes of vector bundles. However, these are…
Brenin
  • 14,072
1 2 3
…
99 100