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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 in coordinate ring

Let $x_{1},x_{2},...,x_{m}$ be elements of $\mathbb{A}^{n}$, where $\mathbb{A}^{n}$ is the n-affine space over an algebraically closed field $k$. Now define $X=\{x_{1},x_{2},...,x_{m}\}$. Why is the coordinate ring $A(X)$, isomorphic to…
user6495
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product of smooth quasi-projective varieties

I want to show that the product of smooth quasi-projective varieties is smooth. Call my varieties $A$ and $B$. Since both are smooth, at every point the dimension of the tangent space equals the dimension of the variety. Will the dimension of $A…
user7461
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Automorphism of a reduced scheme determined by its restriction to a dense open?

Let $X$ be a reduced scheme, and let $U\subset X$ be a dense open, and let $Aut_U(X)$ be the group of automorphisms of $X$ leaving $U$ invariant (ie, which restricts to an action on $U$). Is the natural map $Aut_U(X)\rightarrow Aut(U)$ injective?
user355183
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Morphism between curves constant of surjective

I have a question about a step in the proof of II.6.8 in Hartshorn's "Algebraic Geometry": How he concludes from the fact that $f(X)$ is closed in Y, proper over $k$ and irreducible that $f(X)=Y$ or a point.
user267839
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The Segre embedding is an isomorphism ( Gathmann, Proposition 7.11)

Proposition 7.11(b) (Andreas Gathmann) The Segre embedding $f : \mathbb{P}^m \times \mathbb{P}^n \rightarrow f(\mathbb{P}^m \times \mathbb{P}^n) \subseteq \mathbb{P}^N$ is an isomorphism. Is it an isomorphism of ringed spaces? How can I check? I…
Carlos Albuquerque
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Let $V=\mathcal{V}(XT-YZ)\subset \mathbb{A}^4$. Let $f=\frac{X}{Y}\in k(V)$. What is the domain dom $f$?

I have determined $$A=\{(X,Y,Z,T)\in V\mid Y\neq 0 \textrm{ or } T\neq 0\}\subset \textrm{dom }f$$ However I am unsure of how to show that $A=\textrm{dom }f$.
zerothedog
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Finite group acting on a curve and a line bundle with $G$ action

Let $C$ be a smooth projective curve over $\mathbb{C}$ and $G$ be a finite group acting on $C$. Consider the quotient $f:C\rightarrow C'=C/G$. Suppose that $C'$ is smooth (I think this will be true, since the quotient by a finite group is normal,…
user52991
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Induced Module structure of the Sheaf of Ideals with application to the Sheaf of Relative Differentials

Let $Y$ be a closed subscheme of the scheme $X$ and let $i : Y \rightarrow X$ be the inclusion morphism. Then the sheaf of ideals of $Y$ is defined to be the kernel of the morphism of sheafs $i^{\#}: \mathcal{O}_X \rightarrow i_* \mathcal{O}_Y$.…
Manos
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complex matrices of rank not greater than k is irreducible algebraic variety

So, help guys. How to prove that the set of all complex n×n matrices of rank not greater than $k$ is an irreducible algebraic variety of dimension $k(2n − k)$. Some definitions here: $A$ algebraic variety, if set of its points are common zeros of…
duncan
  • 395
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Elementary question about coordinate ring

I'm trying to learn by myself some algebraic geometry, since I will probably be taking an elementary course next semester. I have some trouble understanding the following: Let $Y=Z(y-x^{2})$ then by definition $Y$ is algebraic set in…
user6495
  • 3,957
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If $G$ is a finite cyclic group acting faithfully on $\mathbb{P}^1_\mathbb{C}$, must it have exactly two fixed points?

If $G$ is a finite (nontrivial) cyclic group acting faithfully on $\mathbb{P}^1_\mathbb{C}$ by holomorphic automorphisms, must it have exactly two fixed points? I believe this should follow from the Hurwitz formula, but it's possible I've made a…
user355183
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How to extend a generic isomorphism to an isomorphism outside finitely many primes?

Let $X,X'$ be normal, proper and flat $\mathbb{Z}$-schemes. Write $X_{\mathbb{Q}}$ for the generic fiber of $X$, i.e. $X_{\mathbb{Q}}:=X \times_{Spec (\mathbb{Z})} Spec (\mathbb{Q})$. Let us assume that there is an isomorphism $X_{\mathbb{Q}}\cong…
Nils Matthes
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When is the canonical sheaf of a curve very ample?

Let $X/k$ be a smooth projective curve over an algebraically closed field $k$ of genus $g$, then when is it that the canonical sheaf $\omega_X$ is very ample, i.e. $\omega_X = i^*\mathcal{O}_{\mathbb{P}^n}(1)$ for some closed immersion…
Dominic Wynter
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Hartshorne Corollary 9.4 precise meaning of $\mathcal{F}_y$ and $\mathcal{F} \otimes k(y)$

Let $f: X \to Y= \operatorname{Spec A}$ be a separated morphism of finite type of noetherian schemes and let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. For a point $y \in Y$, let $X_y$ be the fiber over $y$ and let $\mathcal{F}_y$ be the…
user7090
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Ample invertible sheaves on projective surfaces

This question popped up as I was thinking about intersection theory on fibered surfaces, which is slightly different from the situation I'm about to describe, and as a result I may have gotten some facts wrong. Anyway, here goes: Let $X$ be a…
user355183
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