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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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Intuition for ample line bundles

Let $X\subset \mathbb{P}^N$ be a smooth projective variety over $\mathbb{C}$. We let $\mathcal{O}_X(n)$ denote the bundle induced by $\mathcal{O}_{\mathbb{P}^N}(n)$. For a coherent sheaf $F$ on $X$, we write $F(n):=F\otimes \mathcal{O}_X(n)$. A…
Fermion
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Conics over fields of characteristic two

I was skimming through my solutions of the exercises in Chapter I of Hartshorne and I found two exercises I haven't been able to fully solve. Both exercises are about conics. The first exercise (1.1 c) asks the following: Given an irreducible…
JKaerts
  • 155
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Is Noetherian condition always needed when speaking of a coherent sheaf?

To be precise, I want to strengthen the second part of Proposition 5.4 Chapter 2 in Hartshorne GTM 52 as follows: Let $X$ be a sheme, then an $\mathcal{O}_X$-module $\mathscr{F}$ is coherent if and only if for every open affine subset $U=SpecA$ of…
Li Zhan
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Conditions on Hartshorne exercise II.7.1

Hartshorne, "Algebraic Geometry," Exercise II.7.1, reads: Let $(X, \mathcal{O}_X)$ be a locally ringed space, and let $f : \mathscr{L} \to \mathscr{M}$ be a surjective map of invertible sheaves on $X$. Show that $f$ is an isomorphism. To prove…
Daniel McLaury
  • 24,656
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Problem on quasi - compact morphisms of schemes

I am doing a problem a problem in Hartshorne (2.3.2) which asks to show that a morphism of schemes $f : X \to Y$ is quasi compact iff for every affine open $U \subseteq Y$, $f^{-1}(U)$ is quasi compact. Now one direction is tautological so for the…
user38268
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Reduced schemes and global sections

I am doing exercise 2.2.3(b) in Hartshorne which asks to show that for any scheme $(X,\mathcal{O}_X)$, we have that $(X,({\mathcal{O}_X})_{red}^+)$ is a reduced scheme. By $({\mathcal{O}_X})_{red}$ I mean the the presheaf $U \mapsto…
user38268
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Why are projective morphisms closed?

It is a well-known fact that if $X$ is a projective curve and $p \in X$ a smooth point, then any rational map $X \to Y$, $Y$ a projective variety, extends to a rational map $X \to Y$ regular at $p$. This is proposition I.6.8 in Hartshorne (in the…
Zhen Lin
  • 90,111
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Canonical sheaf not globally generated for a certain surface.

Me and a friend tried the following problem, but with no luck. Anything would be appreciated: Let $X \rightarrow S$ be an arithmetic surface such that for some $s \in S$, $X_s$ is the union of two elliptic curves meeting transversally at a point…
Heidar Svan
  • 411
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Understanding Bertini's theorem

Let's suppose that I am given a pencil generated by the vector fields $X$ and $Y$ in $\mathbb{C}^2$, $\{ Z_\lambda \}_{\lambda\in\mathbb{P}^1}$, that is, $$ Z_\lambda = X + \lambda Y $$ Assume that $X$ and $Y$ are non-singular except in the origin…
OhMyGod
  • 509
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Possible mistake in exercise in Hartshorne exercise II.2.18b

I'm trying to solve Exercise II.2.18b in Hartshorne, and I've constructed what appears to be a counterexample to its statement. Can someone tell me where I've gone wrong? The statement is as follows. Let $\phi : A \rightarrow B$ be a ring…
Jean
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Finite etale maps to the line minus the origin

I am trying to determine the etale fundamental group of $V = A^1 - \{0\}$ over an algebraically closed field $k$. I am trying to stay in the comfortable zone of non-singular varieties. To do this, I wonder if there is an easy way to determine all…
the L
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Is the join of two irreducible varieties is irreducible? (reference + real fied)

The following definition and theorem are taken from J.M. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics, v. 128 (p. 118) Definition 5.1.1.1: The $join$ of two varieties $X$, $Y$…
Ignat Domanov
  • 1,123
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For a (not necessarily affine) scheme $X = \bigcup_{i=1}^n X_{f_i}$, does $(f_1, \ldots, f_n) = (1)$ in $\mathcal{O}_X(X)$?

This is of course true in the affine case, so it seems like it should be true in general, because $\mathcal{O}_X(X)$ should be "smaller" for a non-affine scheme than for a similar affine scheme (e.g. all global sections over a projective scheme are…
Daniel McLaury
  • 24,656
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Rational Curves

I have a set of small questions about rational curves. A rational curve is a curve birationally equivalent to $\mathbb{P}^1$. But I've also seen it said that "$X$ contains a rational curve" when there is a nonconstant (rational?) morphism $f$ from…
user64480
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Degenerations of $\mathbb{P}^1$

Let $X$ be a normal, projective surface (I'm happy to work over $\mathbb{C}$) and let $p: X \to S$ be a flat morphism to a smooth projective curve $S$. Suppose that for a dense open subset of $s \in S$, the fibers $X_s$ are isomorphic to…
Akhil Mathew
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