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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.

29074 questions
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algebraic/geometric interpretation of the projective closure of an affine variety

Let $Y=Z(I)$ be an affine variety of $A^n$ with vanishing ideal $I$. Then $Y$ can be identified with an open set of the projective space $P^n$ and its projective closure $\bar{Y}$ is the closure of $Y$ in $P^n$. What is the algebraic/geometric…
Manos
  • 25,833
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Pullback of Cartier divisors

Let $f:X \to Y$ be a morphism of schemes (not necessarily reduced). Assume $X$ is smooth and $f$ is locally a homeomorphism. Let $D$ be a Cartier divisor on $Y$. We know that the pullback $f^*(D)$ is again a Cartier divisor. If $f^*(D)$ is effective…
Chen
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Functor of order $n$ as in Mumford's Abelian varieties

In section II.6 "the theorem of the cube I" of Mumford's "Abelian Variety" book, Mumford introduced the notion of functor of order $n$. Here is part of the remark immediately below the statement of theorem. Let $T$ be a contravariant functor on…
Jiangwei Xue
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some question of hartshorne book Proposition II (7.3)

Let $k$ be an algebraically closed field, let $X$ be a projective scheme over $k$ and let $f:X\rightarrow \mathbb{P}^n_k$ be a morphism over $k$. Let $\mathcal{L}$ be an invertible sheaf generated global sections $s_0, \dotsc s_n $ such that…
Sang Cheol Lee
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Prove $\mathbb{P}_k^1 \cong \operatorname{Proj} k[x,y,z] / (x^2 + y^2 -z^2)$

Let $k$ be a field with $\operatorname{char} k \neq 2$. I think it's true that: $$\def\Proj{\operatorname{Proj}}\Proj k[x,y,z] / (x^2 + y^2 -z^2) \cong \Proj k[\lambda, \mu] = \mathbb{P}_k^1$$ The left hand side is the projective version of a…
Johann Birnick
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What are the differences among an affine variety, a vector space, and a projective variety?

What are the differences among an affine variety, a vector space, and a projective variety? Are there some nice examples to explain this? Edit: For example, what is the difference between the following: $k^n$ ($k$ is the ground field) as a vector…
LJR
  • 14,520
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Isomorphic varieties

I just want to see if my approach for this problem is fine: Show $W=\mathbb{P}^1 \times \mathbb{P}^1$ is not isomorphic to $W'=\mathbb{P}^2.$ Well $V= \{ [0:1] \} \times \mathbb{P}^1, V' = \{ [1:0] \} \times \mathbb{P}^1$ are closed subvarieties…
Craig
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Base step of induction in Hartshorne III 9.5

Let $f\colon X \to Y$ be a flat morphism of schemes of finite type over a field k. For any point $x\in X$, let $y = f(x)$. Assume that $\operatorname{dim} Y = 0$. Hartshorne now claims that $X_y$ is defined by a nilpotent ideal in $X$. Why? I made…
Rodrigo
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Proving a morphism is étale

I'd like some help to prove that a morphism of schemes $f:X\to Y$ if étale. Here are the characters: $X=\textrm{Spec}\,k[x,x^{-1}]$, $Y=\textrm{Spec}\,k[t]$ and $f$ is induced by $t\mapsto x^2$. [We may assume $k=\overline k$] I tried to show $f$…
Jack
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Map between Zariski tangent spaces(?)

Let $f:X\to Y$ be a morphism of schems, and $x\mapsto y:=f(x)$. Then we have a canonical map $\Phi:T_{X,x}\to T_{Y,y}\otimes_{k(y)}k(x)$ where $T_{X,x}:=\left( \mathfrak{m}_x/\mathfrak{m}_x^2\right)^{\vee}$(dual space) and $k(x)$ is the residue…
User0829
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Section rings map

Let $X$ be a projective variety contained in $\mathbb{P}^n$, over the field of complex numbers. I think that the inclusion $X\subset \mathbb{P}^n$ is controvariantly given by the surjective map among finitely generated…
user618650
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Vector space structure of the Zariski tangent space

Let $X$ be a $k$-scheme and $p\in X(k)$. The Zariski tangent space $T_p X$ is usually defined as being the $k$-vector space $\hom_k(\mathfrak{m}_p/\mathfrak{m}_p^2,k)$. In general, this coincides as a set with $$\widetilde{T_p X}:=\{f\in…
Gabriel
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Why do we use divisors in algebraic geometry?

In algebraic geometry there is a correspondence between Weil divisors, Cartier divisors and line bundles. Over an integral separated locally factorial noetherian scheme, the group of isomorphism classes of Weil divisors and the group of isomorphism…
Gabriel
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What is the idea behind sheaves of rings on distinguished open sets

In the book on algebraic geometry of Mumford (which can be found here), he said : We want to enlarge the ring $R$ into a whole sheaves of rings on $SpecR$, written $\mathcal{O}_{SpecR}=\mathcal{O}_{X}$. So he need to define $\mathcal{O}_{X}(U)$…
Arsenaler
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Is affine $\mathscr O$-connected morphism an isomorphism?

If $\pi :C\to C'$ is an affine $\mathscr O$-connected morphism,then by definition,pull back map $\mathscr O_{C'}(U)\to \mathscr O_C(\pi^{-1}(U))$ is an isomorphism for every affine $U$ of $C'$,since $\pi^{-1}(U)$ is affine,this means…
schuming
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