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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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Isomorphism of coordinate ring of smooth quadric and another subring of a polynomial ring.

I'm looking at exercise 1.8(c) in David Cox' algebraic geometry notes which goes: Let $V=\mathbf{V}(xy-zw) \subset \mathbb{C}^4$. Prove that $\mathbb{C}[V]\cong\mathbb{C}[ab,cd,ac,bd]\subset \mathbb{C}[a,b,c,d]$. I know that…
Billy O.
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Significance of Riemann Roch theorem

In Ravi Vakil's book, the Riemann-Roch theorem for an invertible sheave $\mathscr L$ on regular projective curve $C$ is stated as $$ h^0(C, \mathscr L) - h^1(C, \mathscr L) = deg(\mathscr L) - g +1 $$ where $g$ is the genus of the curve. To me,…
grok
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Is any quasi-projective variety isomorphic to a closed subvariety of a product of a projective space and an affine space?

Let $k$ be an algebraically closed field. Let $\mathbb{P}^n$ be a projective space over $k$. Let $\mathbb{A}^m$ be an affine space over $k$. Is any quasi-projective variety isomorphic to a closed subvariety of $\mathbb{P}^n\times \mathbb{A}^m$ for…
Makoto Kato
  • 42,602
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Schemes covered by finitely-many affine open subsets

Let $X$ be a scheme covered by a finite number of affine open subsets $U_i$ such that for any $U_i, U_j$, the $U_i\cap U_j$ is a union of finite number of affine open subsets $W^{(i,j)}_h$. Then for any affine open subsets $U, V$, the $U\cap V$ is…
Tom
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Is the product of two open embeddings of schemes an open embedding?

Question as in title. I only really need the special case where one of the open embeddings is the identity, but the more general case would be useful to know. Edit - By product I mean: given $U\to X$ and $V\to Y$ open embeddings, is the canonical…
theHigherGeometer
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Hilbert nullstellensatz for schemes

I know that there is a bijection between the irreducible closed sets and the prime ideals of a ring (via functions $V()$ and $I()$). Is this true not only for affine schemes but also for general schemes? And Why?
Tom
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Proof that a Zariski closed subset of a product of projective space is the common zeros of multihomogeneous polynomials

Let $K$ be an algebraically closed field. Let $n, m \ge 0$ be integers. A polynomial $F \in K[x_0,\dots,x_n,y_0,\dots,y_m]$ is called bihomogeneous of bidegree $(p,q)$ if $F$ is a homogeneous polynomial of degree $p$(resp. $q$) when considered as a…
Makoto Kato
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First Chow group of the product of two curves

Scholl's expository paper "Classical Motives" cites Weil's "Sur les courbes algebriques et les varieties qui s'en deduisent," which I have no access to, for the following result: If $X$ and $X'$ are smooth projective curves with Jacobian varieties…
Nehsb
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The image of the d-uple embedding

I have three questions of increasing generality that relate to an exercise in Chapter 1, Section 3 of Hartshorne's Algebraic Geometry concerning the $d$-uple embedding. I'd be happy to have any of them answered. $k$ is an algebraically closed…
Feryll
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What is the fiber of an etale map over a point?

Given a field $K$, $K$-schemes $X$ and $Y$, an étale map $f:X\to Y$, $x\in X$ a point of the topological space to $X$ and $y:=f(x)\in Y$ a point of the topological space to $Y$. What is the basechange $X\times_Y Spec(k(y))$ of $f$ along…
etaleglobale
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What kind of ring $R$ makes $\mathcal O(\mathrm{Proj}\,R[x_0,\cdots,x_n])=R$?

Consider the ring $\Bbb Z[x_0,\cdots,x_n]$ with the grading defined by the degrees of polynomials, then we have $\mathcal O(\mathrm{Proj}\,\Bbb Z[x_0,\cdots,x_n])=\Bbb Z$. Let $R$ be a nonzero commutative ring and consider the ring…
Born to be proud
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Question of Hartshorne book's Proposion II.(2.6)

In Proposition II. (2.6) of Hartshorne book "algebraic geometry". I can't understand proof of part. In proof, let $V$ be an affine variety over field $k$ with the sheaf of regular function $\mathcal{O}_V$ and affine coordinate ring $A$. Let $X={\rm…
Sang Cheol Lee
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Blowing up at a subvariety

Let $Y\subseteq\mathbb{A}^n$ be an affine variety with $\mathbb{I}(Y)=(f_{1},\ldots,f_{s}) \subseteq k[x_{1},\ldots,x_{n}]$. Define $\psi:\mathbb{A}^n \to \mathbb{P}^{s-1}$ by $\psi=(f_{1},\ldots,f_{s})$. And let $\Gamma$ be the graph of $\psi$ in…
user45955
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Tangent space of a point of an affine variety

Let $k$ be a field. Let $f_1,\dots f_r$ be polynomials in $k[x_1,\dots,x_n]$. Let $V$ be an affine variety in $k^n$ defined by $f_1,\dots f_r$, i.e. $V = \{p \in k^n| f_i(p) = 0$ for all $i\}$. Let $p = (a_1,\dots,a_n)$ be a point of $V$. Let $L_i$…
Makoto Kato
  • 42,602
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1 answer

Derive the weak Nullstellensatz from the strong one

I'd like to derive the weak Nullstellensatz An ideal $J\subset K[x_1,\dots,x_n]$ has a common zero exactly if it is a proper ideal. from the strong one $\sqrt{J} = I(V(J))$ This seems pretty easy: \begin{align} J \text{ has no common zero} &…
Andy Miles
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