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.

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Prerequisites for Algebraic Geometry

Is it possible to get into algebraic geometry by just knowing calculus and linear algebra or is this too far of a stretch? If not could anyone give me a list of book/lecture notes recommendations in chronological order to dive into algebraic…
mannequin
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Can every variety appear as singular locus?

let $V$ be a projective variety. Does there always exist another variety $X$ such that $\operatorname{Sing}X=V$? Here $\operatorname{Sing}X$ means the singular locus of $X$ with reduced structure.
Saberization
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Finding singular points and computing dimension of tangent spaces (only for the brave)

I'm currently looking at the following two questions: i) Consider $V = Z(I) \subset \mathbb A_k^3$ where $I$ is generated by $X_1^3 - X_3$ and $X_2^2-X_3$. Find the points at which $V$ is singular and compute the dimensions of the tangent spaces…
Jonathan
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Connected components of a scheme are irreducible

Update 2: I posted an answer to this question. Update 1: Problem is now solved because of the excellent hint by Qil. So, if someone wants to post an answer just for the sake of closing this question you are more than welcome. I will post an answer…
Rankeya
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Why are projective spaces and varieties preferable?

I am reading Hartshorne's Algebraic Geometry and it seems to me that projective spaces and varieties are prefferable. I don't know why. In a more elementary stage of mathematics, when we try to find solutions to given equations, in $\mathbb R$ or…
sunkist
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Polynomials vanishing on an infinite set

I'd like some help making this argument complete and rigorous (if it's correct - if not, help with that would be nice). Here $k$ is a field. Let $A_1,\ldots,A_n \subseteq k$ be infinite subsets. Then any polynomial in $k[x_1,\ldots,x_n]$ that…
smackcrane
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Quasiseparated if finitely covered by affines in appropriate way

I've been reading Vakil's notes on algebraic geometry (on my own -- this is not part of a class), and I'm stuck on one problem (number 6.1.H). It goes as follows. Let $X$ be a scheme. Prove that $X$ is quasicompact and quasiseparated if and only…
T_P
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Hartshorne II prop 6.9

I feel completely in the dark, like I am totally missing what is going on behind the scenes in this section. I apologize in advance. Prop. 6.9: Let $X \to Y$ be a finite morphism of non-singular curves, then for any divisor $D$ on $Y$ we have…
Seth
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Hartshorne II prop 6.6

I'm having a really hard time understanding the proof of this proposition. $X$ is a noetherian integral separated scheme that is regular in codimension 1. We consider $X\times \mathbb{A}^1$ and the projection $\pi$ onto $X$. First, Hartshorne…
Seth
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Definition of "point at infinity"

When looking for integral solutions of Diophantine equations there are sometimes trivial solutions. For example, in the Fermat equation $x^n+y^n=z^n$ such a solution is (1,0,1) and in the cubic equation $x^3+y^3=60z^3$ a trivial solution is…
Ralph
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Cohomology of a tensor product of sheaves

Say I have two locally free sheaves $F,G$ on projective variety $X$. I know the cohomology groups $H^i(X,F)$ and $H^i(X,G)$. Is this enough to give me information about $H^i(X,F\otimes G)$? In particular, if $H^i(X,F)=0$, what conditions on $G$…
Laurent S
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Explanation of "generic point" with examples?

Could someone please explain to me why $X$ and $Y$ are generic points of $\mathbb{R}[X, Y]/(XY)$? And why is the ideal generated by irreducible polynomial is a generic point in $\mathbb{R}[X, Y]$?
Name
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Projective Noether normalization?

In commutative algebra the classic Noether normalization lemma says that every ring finitely generated over a field is a finitely generated module over a polynomial ring with coefficients in this field. The geometric interpretation of this statement…
user115940
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negative multiple of ample line bundle has no global section

Let $X$ be a variety over algebraically closed field $k$, $\dim X > 0$, and let $H$ be ample Cartier divisor on X. Suppose $m < 0$. Show that $\mathcal{O}(mH)$ has no global sections. My result so far is that if $X$ is a complete non-singular curve,…
xyzzyz
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Trying to understand open (closed) subfunctors

I am trying to read about functor of points and I am struggling with the definition of open subfunctor. The definition is the following. A subfunctor $\alpha\colon G\to F$, where $F,G\in \mathsf{Fct}(\mathsf{Rings},\mathsf{Sets})$ is called open…
Sasha Patotski
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