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
6
votes
0 answers

Correspondence to Parameter Space, but not Moduli Space?

I've been thinking recently about some nice moduli problems in algebraic geometry as well as the relationship of moduli spaces to string theory, gauge theory, and such. Mathematically, to my understanding, we first find some parameter space for…
Benighted
  • 2,553
  • 10
  • 23
6
votes
1 answer

functoriality of blow-ups

Let $f:X\to Y$ be a finite map of varieties and let $BL_Z(Y)$ be the blow-up of a subscheme $Z\subset Y$. Is there a map $$\phi:BL_{f^{-1}(Z)}(X)\to BL_{Z}(Y)?$$ If so, what can be said about $\phi$? Is it also finite?
Bonanza
  • 1,203
6
votes
1 answer

Fields of characteristics and affine conics

Let $k$ be any field of characteristic not equal to $2$, and $V$ a $3$-dimensional $k$-vector space. Let $Q: V \to k$, be a non degenerate quadratic form on $V$. How can we show that if $\,0\neq e_1\in V\,$ satisfies $Q(e_1) = 0$, then $V$ has a…
mary
  • 2,374
6
votes
2 answers

Locally free sheaf of rank n on $A^1_k$ is trivial of rank n

The question is how to prove the title, this is actually exercise 13.2.C. from Vakil's notes. The hint is to use the structure theorem for f.g. module over PID. Since to be quasi-coherent is a local property and locally free sheaf is quasi-coherent,…
user330928
  • 826
6
votes
1 answer

Can I choose $k+1$ hypersurfaces to avoid a fiber of dimension $k$ in projective space?

Let $X$ be a closed subscheme of dimension $k$ in $\mathbb{P}^n_A$, where $A$ is a Noetherian ring. In Exercise 11.3.C of Ravi Vakil's notes, it is shown that one may choose $k+1$ hypersurfaces such that the intersection of these hypersurfaces…
Garnet
  • 986
6
votes
1 answer

Pull back of line bundles between projective schemes

Suppose $\phi:S_\bullet \rightarrow R_\bullet$ is a morphism of graded rings that has degree $d$, i.e. $\phi$ maps $S_n$ to $R_{dn}$ for all $n$. Then $\phi$ induces a morphism \begin{equation} \Phi:\text{Proj}~R_\bullet \setminus V(\phi(S_+))…
Wenzhe
  • 2,687
6
votes
1 answer

Surjectivity of linear map between "naive" and "abstract" Zariski tangent spaces

I know this is probably a simple problem but I have got myself stuck trying to prove this fact myself. I'd be very grateful if anyone could clear this up for me. Let $V$ be an affine variety in $\mathbb{A}^n$ (say, over an algebraically closed field…
Alex Saad
  • 3,509
6
votes
1 answer

Parametrising the Clebsch cubic

The Clebsch cubic surface is a famous cubic surface whose 27 lines are all defined over the real numbers. It can be described as the solution set of the equation (in homogeneous coordinates) $$x^3+y^3+z^3+w^3 =(x+y+z+w)^3 \quad (\ast)$$ On the other…
Nefertiti
  • 1,151
6
votes
0 answers

Are Hartshorne's projective morphisms local on the target?

We say that a morphism $f : X \to Y$ is projective if it factors as a closed embedding $i : X \to \mathbb{P}^N_Y$, followed by the projection from $\mathbb{P}^N_Y \to Y$. Question: Is this property local on the target? (I know that the more general…
Elle Najt
  • 20,740
6
votes
1 answer

Example of non-noetherian ring whose spectrum is noetherian

Since spectrum of noetherian ring is a noetherian topological space, I am finding an example s.t. a non-noetherian ring whose spectrum is noetherian. Since most nice rings are noetherian, actually I do not have many examples to start, does any one…
user198206
6
votes
2 answers

Relation between defining polynomials and irreducible components of variety

I've been puzzled about some basic facts in (classical) algebraic geometry, but I cannot seem to find the answer immediately: Let $V=V(f_1,\ldots,f_n)$ be a variety over some field $k$, and let $n > 1$. Suppose that $V$ turned out to be reducible,…
Henry Yuen
  • 693
6
votes
1 answer

How to get the connectedness theorem from the quasi-finite version of ZMT?

Let $f: X \to Y$ be a proper morphism of noetherian schemes. If the natural map $\mathcal{O}_Y \to f_*(\mathcal{O}_X)$ is an isomorphism, then a version of Zariski's main theorem states that the fibers $X_y, y \in Y$ are all connected. (The case…
Akhil Mathew
  • 31,310
6
votes
1 answer

Does $H^i(X,F)\cong H^i(Y,f_*F)$ hold for $X\to Y$ finite but $F$ not necessarily quasi-coherent?

Let $X\to Y$ be a finite morphism between schemes,$F$ be a sheaf of abelian groups but not necesarily quasi-coherent. Does $H^i(X,F)\cong H^i(Y,f_*F)$ still hold for sheaf cohomology with Zariski topology? (It holds for etale cohomology)
user93417
6
votes
2 answers

The motivation of stability via Mumford slopes

A vector bundle $F$ is called (semi)stable if Mumford slopes $\mu=c_1 / rk$ of all the subbundles are less (or equal) then Mumford slope of $F$. Can you explain the motivation of this definition or give me a good reference?
evgeny
  • 3,781
6
votes
2 answers

quasi-finite maps to quasi-projective varieties?

Say we are given a smooth complex algebraic variety $Y$ which is quasi-projective, and $X$ a second complex manifold together with a holomorphic map $f:X\rightarrow Y$ which is of finite fibers, is it true that $X$ remains quasi-projective? Is the…
turtle
  • 163
  • 3
1 2 3
…
99 100