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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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Notation: $\mathbb{P}(\mathscr{E})$, where $\mathscr{E}$ is locally free sheaf.

Let $\mathscr{E}$ be a locally free sheaf on a scheme $X$. I always thought that $\mathbb{P}(\mathscr{E})$ meant $\mathrm{\bf Proj}(\textrm{Sym}(\mathscr{E}))$, the global Proj associated to the sheaf of symmetric algebra of $\mathscr{E}$. But, when…
rla
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Is a reducible variety projective if each of its irreducible component is?

Suppose $X=X_1\cup\cdots\cup X_n$ is an irreducible decomposition of variety(scheme). Suppose each $X_i$ is projective, then is $X$ projective? Does this hold for proper case? The local ring at the intersection became unclear to me, I am not sure if…
user93417
3
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Why that a quasi-projective variety being isomorphic to a projective variety must itself be projective as well

On page 54 in the Book An Invitation to Algebraic Geometry, the author said that A quasi-projective variety in $\mathbb{P}^n$ is isomorphic to a Zariski-closed subset of some $\mathbb{P}^m$ (i.e. a projective variety) if and only if it already…
hxhxhx88
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An exercise from the Harris's book

I need a hint on an exercise: Let $K$ be an algebraically closed field. Prove that any finite set $\Gamma \subset KP^n$ such that not all of its points lie on the same line can be given by polynomials of degree less than…
Aleksei Averchenko
  • 8,023
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$f_*(O_X)=O_Y$ and connectedness of fibers

Suppose $X\to Y $ is a morphism , under what conditions we have direct image sheaf $f_*(O_X)=O_Y$? For example, suppose $\tilde{S}\to S$ is a blow up, do we have $f_*(O_{\tilde{S}})=O_S$? Hartshorne III 11.3 says that $f_*(O_X)=O_Y$ implies the…
user93417
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1 answer

étale fundamental group of strictly henselian discrete valuation ring minus closed point

Let $A$ be a strictly henselian discrete valuation ring. What is $\pi_1(\operatorname{Spec}(A) \setminus \{s\})$? I thought it is a semidirect product of a pro-$p$-group (the wild ramification group) and $\hat{\mathbf{Z}}'(1)$ (the tame fundamental…
user5262
  • 1,863
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Two continuous functions agree on an open subset of an irreducible space.

I have the varieties $X,Y,Z$ where $X$ is complete. I have the morphism of varieties $f:X\times Y\rightarrow Z$, I have a closed subset $W\subset Y$ such that $f=g\circ\textrm{pr}_Y$ on $X\times(Y\setminus W)$ where $g(y)=f(x_0,y)$ for some $x_0\in…
user99412
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When does a field extension canonically determine a morphism of schemes?

If I have an extension $L/K$ of number fields, then I can take the inclusion $\mathcal{O}_K \hookrightarrow \mathcal{O}_L$ and get a morphism of "curves" $\operatorname{Spec} \mathcal{O}_L \to \operatorname{Spec} \mathcal{O}_K$. Can I do something…
Daniel McLaury
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Do rational functions separate points?

Let $X$ be an irreducible, normal variety over an algebraically closed field of characteristic zero. Let $x,y\in X$ be two points such that $f(x)=f(y)$ for every $f\in K(X)$ which is defined at $x$ and at $y$. Can I conclude that $x=y$? I feel the…
Jesko Hüttenhain
  • 14,792
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1 answer

How to determine the local ring

In general, how does one determine a local ring. And in particular, how would one do it for $O_{A}(A $ \ $ \{(0)\})$, where A is 1-dim affine space in $\mathbb{C}$?
math1234567
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Spectral sequence differentials

Let $\mathcal{F}^{\bullet}$ and $\mathcal{G}^{\bullet}$ be complexes of coherent sheaves on a variety $X$. There is a spectral sequence $$E_2^{p,q}=\mathcal{Ext}^p(\mathcal{H}^q(\mathcal{F}^{\bullet}),\mathcal{G}^{\bullet}) \Longrightarrow…
Luc
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Resolution of an affine variety

Did exists a method to find a resolution of singularities of an affine algebraic variety over $\mathbb{C}$ such that its resolution is affine too?
Joaquín Moraga
  • 423
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Hartshorne Lemma ( I 6.4 )

I have difficulty to understand the proof of this lemma : Lemma 6.4 Hartshorne Let $Y$ be a qausi-projective variety, let $P,Q\in Y,$ and suppose that $\mathcal{O}_P\subset\mathcal{O}_Q$ as subrings of $K(Y).$ Then $ P= Q.$ Embed $Y$ in…
Med
  • 1,598
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A criterion for quasiaffine varities

If $X$ is a prevariety and the open sets $X_f:=\{x:f(x)\neq 0\}$ form a basis for the topology of $X$ as $f$ varies over $\Gamma(X,\mathcal{O}_X)$, then why is $X$ quasi-affine? The definitions are from Mumford's red book, so prevariety is a scheme…
DCT
  • 3,315
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Separating tangent vector

Let $X$ be a projective scheme over an algebraically closed field $k$, $\mathcal{L}$ an invetible sheaf on $X$ and $V \subseteq \mathcal{L}(X)$. I saw in Hartshorne's book(p.152) that elements of $V$ separate tangent vectors if for each closed point…
Sang Cheol Lee
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