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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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Linear form does not vanish on any component

I read Shafarevich basic algebraic geometry on page 70 he says for any $X \subseteq \Bbb{P}^N$ can find a linear form $L$ that does not vanish on any component of $X$, I tried to prove this by induction but failed how to prove this?
user23086
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Isomorphism between global sections of normal sheaf and $\mathrm{Ext}^1_X(i_*\mathcal{O}_Z, i_*\mathcal{O}_Z)$

Let $X$ be an n–dimensional compact algebraic manifold, $i \colon Z \to X$ a closed submanifold, and let $\mathcal{N}_{Z|X}$ denote the normal sheaf. I have trouble understanding the isomorphism $$ H^0(Z,\mathcal{N}_{Z|X}) \xrightarrow{\cong}…
Christa Wolf
  • 163
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Chern character

I am trying to read the book “Fourier-Mukai transforms in algebraic geometry”. Around the end of the page 126 of this book, it is written that the Chern character is from $K(X)$ (Grothendieck group) to $H^*(X,\mathbb{Q})$. However, according to…
Pouya Layeghi
  • 1,470
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Matrices of rank and singular points

I am stuck on the following homework problem: Show that if S=S(m,n,r) represents the space of m by n matrices with rank less than or equal to r (naturally isomorphic to an affine subvariety of $\mathbb{A}^{mn}$ of course) over some arbitrary field…
Peter Hewkin
  • 63
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Hartshorne III.7.6b) (ii) => (i) "Duality for a projective scheme)

Let X be a closed immersion of dimension n in P = *P*$^N_k$, where k is an algebraically closed field. Let $\omega_P$ denote the canonical bundle and A the local ring $\mathcal O_{P,x}$. Then Hartshorne argues on p. 244 that the condition $$\mathcal…
Rodrigo
  • 7,646
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Dimension of irreducible of finite type k-scheme

I saw a claim that states for a field $k$ and an irreducible of finite type $k$ scheme $X$, $\textrm{dim}X=\textrm{dim} \mathcal{O}_{X,x}$ for any closed point $x$. The proof starts with reducing the case $X$ is integral affine scheme, but I cannot…
User0829
  • 1,359
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How to tell $\overline{Z(f)\cap V}=V$?

Say $V$ is an irreducible variety in $\mathbb{C}^n$ of dimension $d$. I can view $\mathbb{C}^n$ as $\mathbb{R}^{2n}$ and inside $\mathbb{R}^{2n}$, $V$ is an irreducible (real) variety of dimension $2d$. If I pick a polynomial…
Levent
  • 4,804
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How to imagine a plane defined by Cartesian Plane Equation?

It isn't difficult for me to imagine a plane based on three points. Also it is quite simple to imagine a plane based on point and normal vector. Are there some tricks to imagine a plane defined by plane equation $Ax+By+Cz+D=0?$
kletnoe
  • 133
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Is the closure of the image of $\mathbb C^n$ under vector-polynomial map is irreducible affine variety of dimension not higher than $n$?

I work with the subset $Z\subset\mathbb C^m$ which is the image of $\mathbb C^n$ under vector-polynomial map $f(x)=(p_1(x),...,p_m(x))$, that is $f$ sends $x\in\mathbb C^n$ to $f(x)\in Z\subset\mathbb C^m$. Is the following true? 1) $Z$ is…
Ignat Domanov
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Don't understand proof on pg 65 of Qing Liu

There is proposition in page 65 of Liu's book which is: $X$ an integral scheme with generic point $\xi$. Then if we identify $\mathcal{O}_X(U)$ with and $\mathcal{O}_{X,x}$ we have $\mathcal{O}_X(U) = \bigcap_{x \in U} \mathcal{O}_{X,x}$. His proof…
user23086
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1 answer

Does this morphism extend to projective varieties?

We work over the complex numbers. Let $\mathbb{A}^1\times \mathbb{A}^1\to \mathbb{A}^1$ be the morphism given by $(x,y)\mapsto x-y$. Does this extend to a morphism $\mathbb{P}^1\times \mathbb{P}^1\to \mathbb{P}^1$? I have the feeling it does not…
Hinter
  • 133
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How to complete this proof on the codimension of the intersection of affine varieties

Edit: I think maybe I'm making this way harder than it needs to be. If $L\subsetneq R:= k[z_1,...,z_m]$ is an ideal such that $V(L)$ is equidimensional and $f_1,...,f_p\in R$, then it seems like Krull's PIT gives $$dim(R/\mathfrak{p}) \geq…
dessin d'enfant terrible
  • 503
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Normal Cone as a Quotient Orbit Space

Let $V=V(J)$ be the variety defined by the ideal $J\subset\Bbb{C}[x_0,\dots,x_d]$ generated by the 2x2 minors of $$\begin{pmatrix} x_0 & x_1 &\cdots & x_{d-1} \\ x_1 & x_2 &\cdots & x_d \end{pmatrix}$$ (in other word by the set of equations…
Heitor Fontana
  • 3,676
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How to define the canonical Godement resolution

Good afternoon : I whould like to know how to define the canonical Godement resolution of a flasque sheaf. Thanks a lot.
Bryan
  • 827
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Open and Closed Set in Zariski Topology

I'm confused about the definition closed and open set in Zariski Topology, it is said that the set $$V(I)=\{P \in \operatorname{Spec}(R)\mid I \subseteq P\}$$ are the closed set in Zariski Topology. But it is said in James Munkres's Topology that a…
Blackoffe
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