Mathematics Stack Exchange
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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.

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explicit equations for blowing down

If you have an explicit equation for a (-1) curve on a surface, and an explicit equation for the surface, so that the surface comes as a blowup with the (-1) curve being the exceptional divisor, is there some formula for writing down the equations…
Tony
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Irreducibility of smooth intersection

In many concrete cases I find it quite hard to show the irreducibility of a given variety. Can I proceed in the following way: Situation: We look at the closed subscheme $$X = V(f_1, \dots, f_n) \subset \mathbb{P}^m_k.$$ for a algebraically closed…
Casaubon
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Is prime spectrum the soberification of maximal spectrum?

Given a commutative ring $R$, let $\operatorname{Spec} R$ denote its set of prime ideals, equipped with the Zariski topology. Let $\operatorname{Spm} R$ denote the maximal spectrum of $R$, which is the topological subspace of $\operatorname{Spec} R$…
V. Semeria
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A fake proof of pull-back of an ample divisor is ample

Let $\pi: X \to Y$ be a morphism between projective varieties and $H$ be an ample divisor on $Y$. I want to "show" $\pi^*H$ is ample on $X$ by the following argument: Let $N_1(X)_\mathbb{R},N_1(Y)_\mathbb{R}$ be vector spaces of 1-cycles module…
Li Yutong
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Hartshorne IV Exercise 6.9: How to show $Y$ is contained in the hyperplane section?

The exercise is stated as follows: 6.9. Let $X$ be an irreducible nonsingular curve in $\mathbf{P}^{3}$. Then for each $m\gg0$, there is a nonsingular surface $F$ of degree $m$ containing $X$. [Hint: Let $\pi: \tilde{\mathbf{P}} \rightarrow…
Richard
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What led Grothendieck to emphasize the "relative perspective"?

It is often said that one of the most influential concepts Grothendieck introduced through the scheme theory is the emphasis on the "relative perspective," that is, properties should be interpreted as a property of morphisms instead of one of the…
Takuei ASAI
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Algebraic Groups Problem

I am currently trying to go through Humphrey's Linear Algebraic Groups and am stuck on a problem that sounds deceivingly simple but can not seem to figure out (Problem 5 from Section 7). I have spent a fair amount of time on it and expect it to be…
Sean Ballentine
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example of locally finite type not finite type

Let $A$ be a ring then, a homomorphism $A\rightarrow A[x_1,\cdots,x_n]$ induces a finite type morphism between spectrums. I want to find the map which is a locally finite type but not finite type....
Sang Cheol Lee
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How to imagine $X^2+Y^2-1=0$ in $C^2$, $X$, $Y$ both complex

Like the title, with a graph if convenient, I was reading Mumford's book, a picture make me confused: Could someone explain it to me, thanks.
Yoshinobu Osawa
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Is this an affine surface?

I have to prove that the surface $Z(xy^2-z^2)$ in the affine space is a rational one, where the base field is the complex one. Is it right to use the morphism $t \rightarrow (t,t,t^{3/2})$ with the projection over the first component for the…
user52342
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Fibre products of closed subschemes

Let $U = Spec(A)$ and $V = Spec(B)$ be affine schemes. Let $X$ be a separated scheme. Suppose that there exist morphisms $U \to X$ and $V \to X$. Then is the natural map $$A \otimes_{\Gamma(X, \mathcal O_X)} B \to \Gamma(U \times_X V, O_{U…
Carl
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Find the rational parametrization of this curve

I need to find the rational parametrization of the complex curve with affine equation $$y^2-6x^2y-3x^4+4x^3y+4x^3=0.$$ I did find a solution, I don't know if there is a more elegant solution, but this is what I did nonetheless. I took the lines…
Wastaken
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Domain of definition of a rational automorphism

I am a bit confused about the following problem from Liu's book on Algebraic geometry. Let $X= \mathbb{P}^1_\mathbb{Z}$ and let $f: X_\mathbb{Q} \rightarrow X_\mathbb{Q}$ be an automorphism corresponding to a matrix of $PGL_2 (\mathbb{Q})$.…
Heidar Svan
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Castelnuovo-Mumford regularity - Mumford's proof

I am studying Mumford's 'Lectures on curves on an algebraic surface', in particular the section on Castelnuovo Mumford regularity. I am stuck in a small point. A coherent sheaf $F$ on $P^n$ is $m$-regular if $H^i(P^n,F(m-i))=0$ for all $i>0$. He…
user52991
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Quasi-coherent sheaves on punctured affine plane

Let $A=k[x,y]$ be the polynomial ring in two variables over a field $k$, $\mathbb{A}^2=\mathrm{Spec} A$ the affine plane , and $U=\mathbb{A}^2 \setminus \{0\}$ the punctured affine plane. Then the canonical injection $j : U \to \mathbb{A}^2$…
LOCOAS
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