Mathematics Stack Exchange
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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Picard group of product of spaces

Suppose $X,Y$ are varieties over an algebraically closed field $k$. Can we compute $\operatorname{Pic}(X \times_k Y) $ in terms of $\operatorname{Pic}(X),\operatorname{Pic}(Y)$? It seems that $\operatorname{Pic}(X \times_k Y) \cong…
Li Yutong
  • 4,065
23
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2 answers

Does Hom commute with stalks for locally free sheaves?

This is somewhat related to the question Why doesn't Hom commute with taking stalks?. My question is this: If $F$ and $G$ are locally free sheaves of $\mathcal{O}_X$ -modules on an arbitrary ringed space $(X,\mathcal{O}_X)$, then is the stalk of the…
Enrique Acosta
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Closed immersions are stable under base change

My question can be summarized as: I want to prove that closed immersions are stable under base change. This is exercise II.3.11.a in Hartshorne's Algebraic Geometry. I researched this for about half a day. I consulted a number of books and online…
PeterM
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Open subschemes of affine schemes are affine?

I'm reading Hartshorne, and was wondering if this was true: Let $A$ be a ring and let $U$ be an open subset of $Spec(A)$. Let $S$ be the set of elements of $A$ not in any prime of $U$. Then $S$ is multiplicatively closed and $\mathcal{O}_A|_U \cong…
Zach L.
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An open subset of an irreducible set is dense.

I'm trying to understand this example in Hartshorne's algebraic geometry book In order to prove the irreducible part, suppose $Y$ is an irreducible space and $Y'$ a open subset of $Y$ with $Y'=Y'_1\cup Y'_2$ with $Y'_1,Y'_2$ proper closed subsets.…
user42912
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Dominant morphism between affine varieties induces injection on coordinate rings?

Here are the definitions that we use for this problem: A morphism $\varphi : X \to Y$ between two varieties is said to be dominant if the image of $\varphi$ is dense in $Y$ (c.f. Hartshorne exercise 1.3.17) We say $X$ is an affine variety if it is…
user38268
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$\mathbb{A}^2\backslash\{(0,0)\}$ is not affine variety

In our lecture notes we have this example, with the proof why $X = \Bbb{A}^2\setminus \{(0,0)\}$ is not an affine variety: Let $i:X\hookrightarrow \mathbb{A}^2$ be an inclusion map. We show, that any regular function on $X$ extends uniquely to a…
Luca
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Why study schemes?

Why study schemes instead of only affine/projective varieties, given by zeros of polynomials in the affine/projective space? I mean, what is gained by introducing the concept of schemes? Thank you!
rla
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Sard's theorem for algebraic varieties

(One version of) Sard's theorem states that: Theorem (Sard): Given $M$ and $N$ smooth manifolds of dimensions $m$ and $n$ respectively, and a smooth map $f:M\to N$, then the set of singular values of $f$ has measure zero. A corollary of this…
Daniel Robert-Nicoud
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When is a flat morphism open?

Hartshorne, Algebraic Geometry, Exercise III.9.1 asks one to prove A flat morphism $f : X \to Y$ of finite type of Noetherian schemes is open, i.e., for every open subset $U \subseteq X$, $f(U)$ is open in $Y$. So far as I can tell this is…
Daniel McLaury
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3 answers

Does Proj induce some equivalence of categories involving graded rings?

The opposite category of the category of rings is equivalent to the category of affine schemes, via the Spec functor. Is there a similar result if we consider the Proj construction, that takes a graded ring and returns some scheme? This is a…
Bruno Stonek
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What is the definition of surjective morphism of schemes?

Let $f: X \to Y$ be a morphism of schemes, when talking about the surjectivity of $f$, there are at least several possibilities. (1) $f$ is surjective at the level of sets, that is $\forall \ y \in Y$, there exist $x \in X$, such that $f(x)=y$. (2)…
Li Yutong
  • 4,065
19
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2 answers

How did people get the inspiration for the sums of cubes formula?

I stumbled upon this neat formula for sums of cubes with arbitrary $x,y\in\mathbb{Z}$$$(x^2+9xy-y^2)^3+(12x^2-4xy+2y^2)^3=(9x^2-7xy-y^2)^3+(10x^2+2y^2)^3\tag1$$ With $1729=1^3+12^3=9^3+10^3$ as its first instance. And I believe that this formula was…
Frank
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Zariski dense implies classically dense?

I was surprised that I wasn't able to find this question already posted; if it has been posted and I just didn't find the right search terms, let me know. Let $X$ be any complex variety. A priori, any set which is dense in the classical topology on…
Dustan Levenstein
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Difference between a stalk of a sheaf and a fiber of a vector bundle

Is there an analogy between fibers $ \pi^{-1} ( x ) $ of a vector bundle $ \pi : E \to X $, and the stalk $ \mathcal{F}_x $ of a sheaf $ \mathcal{F} $ défined by : $ \mathcal{F}_x = \displaystyle \lim \mathcal{F} ( U ) $ : the direct limit over all…
Bryan
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