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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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Kernel of a morphism from a locally free sheaf is locally free

Let $C$ be a projective curve (not necessarily reduced or irreducible). Let $\mathcal{F}, \mathcal{G}$ be $\mathcal{O}_C$-modules and $\phi:\mathcal{F} \to \mathcal{G}$ be a morphism of $\mathcal{O}_C$-modules. Suppose further that $\mathcal{F}$ is…
Chen
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Glueing morphisms of sheaves together - can I just do this?

While trying to solve a certain exercise in Hartshorne I realized that I need to use the following result: Let $X,Y$ be two ringed topological spaces. Suppose we have a covering $\{U_i\}$ of $X$ and morphisms $f_i :U_i \to Y$ such that $f_i|_{U_i…
user38268
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Two definitions of a projective morphism

In Hartshorne's Algebraic Geometry, page 103, a morphism $f: X \rightarrow Y$ is said to be projective if it factors as a closed immersion $X \rightarrow {\bf P}^n_Y$ followed by the projection ${\bf P}^n_Y \rightarrow Y$. As noted there, EGA II,…
Visitor
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Problem in Rick Miranda: finding genus of a projective curve

I have just started learning Riemann surfaces and I am using the book by Rick Miranda: Algebraic curves and Riemann Surfaces. #F in section 1.3 asks to determine the genus of the curve in $\mathbb{P}^3$ defined by the two equations $x_0x_3=2x_1x_2$…
Divakaran
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An irreducible polynomial $f \in \mathbb R[x,y]$, whose zero set in $\mathbb A_{\mathbb R}^2$ is not irreducible

This is an exercise on Page 8 of Hartshone's Algebraic Geometry: Give an example of an irreducible polynomial $f \in \mathbb R[x,y]$, whose zero set $Z(f)$ in $\mathbb A_{\mathbb R}^2$ is not irreducible. I think such an example must come from the…
ShinyaSakai
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Example of an ample line bundle $L$ with $L^{\otimes m}$ very ample and $L^{\otimes (m+1)}$ not very ample

Let $X$ be a smooth projective variety over $\mathbb{C}$. Is there an ample line bundle $L$ such that $L^{\otimes m}$ is very ample, but $L^{\otimes(m+1)}$ is not very ample? I expect such an $L$ to exist, though I have not been able to construct…
msteve
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Cohomology of quasi-coherent sheaves with respect to pushforward question (Exercise in Hartshorne)

I am confused about something in Exercise 4.1 of Chapter 3 of Hartshorne. It asks us to prove : Let $f : X \rightarrow Y$ be an affine morphism of Noetherian, separated schemes. Show that for any quasi-coherent sheaf $\mathcal{F}$ on $X$, there…
David
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Why are affine varieties except points not compact in the standard topology on $C^n$ ?

I am starting to learn algebraic geometry and in the notes I am reading there is the following remark: " Over the complex numbers and with the strong topology we see that $A^n$ and affine varieties (except for points) are not compact. " (The strong…
readingframe
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What's the intution behind defining the cotangent sheaf as $\Delta^\ast(\mathscr{I}/\mathscr{I}^2)$?

This definition seems to be given all over the place (e.g. Hartshorne II.8, Vakil 21.2.20, Wikipedia, McKernan's lecture notes from MIT), and never with any explanation as to why the map $\Delta : X \to X \times X$ should have any relation to…
Daniel McLaury
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Is skyscraper sheaf quasi-coherent?

Suppose $\mathcal{F}$ is a skyscraper sheaf supported on $\bar{\{\mathfrak{p}\}}$, the stalk is $M$, What is its global section over $\operatorname{Spec} A$? We need to find a module $N$ such that $N_{\mathfrak{q}}=M$ when $\mathfrak{q}\in…
user93417
14
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1 answer

Is a scheme with a single closed point affine?

Let $X$ be a quasi-compact, separated scheme with a single closed point. Is $X$ necessarily affine, and thus isomorphic to the spectrum of a local ring? I could not think of a counter-example; is there one?
Bruno Joyal
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Quasi-separatedness is affine-local?

[Vakil defines a scheme $X$ to be quasiseparated if the intersection of any two quasicompact opens is quasicompact] This is part (b) of 7.3.C in Vakil's FOAG: Show that a morphism $\pi$ from a scheme $X$ into a scheme $Y$ is quasiseparated [ For any…
John
  • 253
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Properties of quotient sheaves

I am reading Hartshorne's Algebraic Geometry, II.6 about Cartier Divisor. It is defined to be the global section of the sheaf $K^*/O^*$. Then it said: " thinking of the properties of the quotient sheaves, we see that a Cartier divisor can be…
Long
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Global sections of the line bundle $\mathcal{O}(D)$

For $D$ a divisor on a smooth variety $X$ over $\mathbb{C}$, we define as usual the subsheaf $\mathcal{O} (D)=\mathcal{O}_X(D)$ of the sheaf of rational functions $\mathcal{K}_X$ as follows: $$\mathcal{O}_X(D)(U):=\{ f\in\mathcal{K}_X(U)\;|\;…
Simplicius
  • 485
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2 answers

Intuition Behind, or Canonical Examples of Finite Type Morphisms

I'm new to the world of schemes, so I'm still trying to grasp some of the basics. I understand most of the simple topological properties of schemes, as well as some of the sheaf-theoretic properties like reducedness, integrality, normality, etc. …
Benighted
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