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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
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Why is the infinite union of algebraic sets not necessarily algebraic?

Let $\left\{Y_i\right\}_{i \in \mathcal{I}}$, where $\mathcal{I}$ is infinite, be a family of algebraic sets of $k^n$, where $k$ is an algebraically closed field. Then $Y_i = \mathcal{Z}(T_i)$, i.e. each $Y_i$ is the zeros of some subset $T_i$ of…
Manos
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Unique rational normal curve through d+3 points

We define a rational normal curve to be the image of a map $$\mathbb P^1\rightarrow \mathbb P^d, [x:y]\mapsto [P_0(x,y):P_1(x,y): \ldots :P_d(x,y)]$$ where $P_0(x,y),P_1(x,y), \ldots P_d(x,y)$ are linearly independent homogeneous degree $d$…
user223794
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Relating the homogeneous coordinate ring of a projective variety with the affine coordinate ring of an affine open subset

I'm currently working on exercise 2.6 in chapter 1 of Algebraic Geometry by Hartshorne. I'm pretty confident with my answer apart from the first bit which I feel I could be "fudging". I'm looking for people to tell me where I have gone wrong... Here…
M Davolo
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What's the difference between $\mathbb{A}^n$ and $\mathbb{A}^{n+1}$?

Besides the obvious difference in topological dimension. If you want to distinguish between $\mathbb{R}$ and $\mathbb{R}^2$, take an open set in the plane, remove a point, then it's still connected. That doesn't work in $\mathbb{R}$. If you want…
D_S
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Is the restriction of a finite map of affine varieties also finite?

If $f:X\to Y$ is a dominant(i.e.$f(X)$ is dense in $Y$) regular map of affine varieties, then $f$ is called a finite map if $k[X]$ is integral over $f^*k[Y]$. My question is: if $Z\subset X$ is a closed subset of $X$, then how to show the…
Hang
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Is there an easy criterion to determine whether given polynomials form a complete intersection?

Suppose we have homogeneous polynomials in $s$ variables $F_1, ..., F_n$ with coefficients in integers. Let $X$ be a variety (or algebraic set) defined by the simultaneous equations $$ F_1(\mathbf{x}) = ... = F_n(\mathbf{x}) = 0. $$ I was…
Johnny T.
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Restriction of flat morphism

Suppose that $f\colon X\to Y$ is a flat morphism of varieties over an algebraically closed field $k$. Let $E\subseteq X$ and $F\subseteq Y$ be closed subvarieties such that $f(E) = F$. Is it true that the restricted morphism $f|_E\colon E\to F$ is…
froggie
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Is rank of vector bundle encoded in its Hilbert polynomial

Let $\mathcal F$ be a vector bundle over a projective variety $(X, \mathcal O_X(1))$, and $P_\mathcal F(m)=\chi(X, \mathcal F(m))$ be its Hilbert polynomial. Then can I define from $P_\mathcal F$ the value of the rank $rk(\mathcal F)$?
evgeny
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Surjective map on coordinate rings implies the map is injective

Let $X$ and $Y$ be affine varieties and $f: X \rightarrow Y$ a polynomial map. If the induced map on coordinate rings $K[Y] \rightarrow K[X]$ is surjective why this implies that $f$ is injective?
user31509
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Projective closure

Is the projective closure of an infinite affine variety (over an algebraically closed field, I only care about the classical case right now) always strictly larger than the affine variety? I know it is an open dense subset of its projective closure,…
Justin Campbell
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Question about Hartshorne's proof of Halphen's Theorem (proposition IV.6.1)

My question comes from the proof of Theorem 6.1 in section IV.6 of Hartshorne, where I don't understand the very last step. The theorem is as follows: A curve $X$ of genus $g\geq 2$ has a nonspecial very ample divisor $D$ of degree $d$ if and only…
Garnet
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Hartshorne's proof of the exact sequence $\mathbb{Z} \to \operatorname{Cl} X \to \operatorname{Cl} U \to 0$

Hartshorne, Algebraic Geometry, Proposition II.6.5 reads (in part): Let $X$ satisfy (*), let $Z$ be a proper closed subset of $X$, and let $U = X \setminus Z$. Then: [...] (c) if $Z$ is an irreducible subset of codimension 1, then there is an…
Daniel McLaury
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Fine moduli space and modular curves

I am a bit confused about the definition of fine moduli spaces. As far as I understand, the difference between fine moduli space and coarse moduli space is the existence of universal family. For fine moduli space, suppose that $U\to M$ is the…
Jiangwei Xue
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Morphism between projective schemes induced by surjection of graded rings

Ravi Vakil 9.2.B is "Suppose that $S \rightarrow R$ is a surjection of graded rings. Show that the induced morphism $\text{Proj }R \rightarrow \text{Proj }S$ is a closed embedding." I don't even see how to prove that the morphism is affine. The…
only
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formal glueing of schemes

I need to know when a glueing prcedure makes sense in the scheme world. Let $X$ be a smooth, projective scheme over $S=Spec(A)$, $A$ a complete ring. Let $x,y\in X$ closed points such that $\phi:\hat{\mathcal{O}}_{X,x}\cong \hat{\mathcal{O}}_{X,y}$…
umino
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