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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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Why are constructible sets a disjoint unions of locally closed sets

Let $X$ be a noetherian scheme. The constructible sets are the smallest boolean algebra containing all of the open sets. It is easy to see that the constructible sets are exactly finite unions of locally closed sets. I have read several times that…
DBr
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Is it true that for algebraic sets $V,W$ we have $I(V \times W ) =I(V) + I(W)$?

This is a follow up question to my previous question here. Let $k$ be a field and $V \subseteq \Bbb{A}^n$ and $W \subseteq \Bbb{A}^m$ be algebraic sets. Then it should be true that $I(V \times W ) = I(V) + I(W)$ where by $I(V)$ here we mean the…
user38268
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2 answers

Elementary question about Cayley Hamilton theorem and Zariski topology

A question about a proof of the Cayley-Hamilton theorem using Zariski topology. "The set $C$ of all matrices of size $n \times n$ (over an algebraically closed field $k$) with distinct eigenvalues is dense in the Zariski topology". Can we argue as…
user6495
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What is the relationship between different theorems all called Hilbert's Nullstellensatz?

The following statements are all named the Hilbert's Nullstellensatz, but they appear at first to be completely unrelated to each other. What is the relationship between them exactly? (Theorem 1.3A on page 4 of Hartshorne's Algebraic Geometry) Let…
David Lui
  • 6,295
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Hartshorne exercise 1.1 (c)

I want to know how can I find the conditions where $A(W)$, the affine coordinate ring of a variety given by an irreducible quadratic polynomial in $k[x,y]$, is isomorphic to $A(V)$ or to $A(Z)$ where $V$ is a parabola defined by $y=x^2$ and $Z$ is…
Rodrigo Torres
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Geometric intuition for flat morphisms

I'm trying to develop some geometric intuition for what it means for a morphism of schemes to be flat. The definition of flatness in Hartshorne says (if I'm correct) that a morphism $f: X \to Y$ is flat iff pullbacks of SESs of quasicoherent sheaves…
Kenny Wong
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Concrete example of calculation of $\ell$-adic cohomology

Let $p$ and $\ell$ be distinct prime numbers. Consider in the affine plane $\mathbb{A}^2_{\mathbb{F}_p}$ with coordinates $(x,y)$ the union $L$ of the axes $x = 0$ and $y = 0$. How does one compute the $\ell$-adic cohomology groups with compact…
Evariste
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Is it possible that the product of two non-affine schemes becomes affine?

Question: Is there an example of some $X$ and $Y$ non-affine schemes, with $X \times_{\operatorname{Spec} \mathbb{Z}} Y$ affine? Updated question (after Eric Wofsey's example): Is there an example of some $X$ and $Y$ non-affine $k$-schemes, with $X…
Elle Najt
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Normal bundle to complete intersection in $\mathbb{P}^n$

Let $X\subset\mathbb{P}^n$ be a complete intersection defined by irreducible polynomials $f_1,...,f_k$ of degrees $d_1,...,d_k$. How to show that the normal bundle of $X$ is isomorphic to $\bigoplus\limits_{i=1}^k\mathcal{O}_X(d_i)$?
mahavishnu
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Why is $F^*(\mathcal L) = \mathcal L^{\otimes p}$ where $F$ is the absolute Frobenius and $\mathcal L$ is an invertible sheaf?

Suppose $S$ is such that $\mathcal O_S$ is killed by multiplication by $p$. The absolute Frobenius $F: S \to S$ is defined to be be the identity on the underlying points topological space of $S$, with the sheaf map $F^\#: \mathcal O_S \to \mathcal…
maxymoo
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Local Ring of a Subvariety (problem 1.3.13 in Hartshorne)

I've began to study AG to prepare for grad school and I'm stuck with the following problem in Hartshorne. The problem statement is as follows (found on p. 22 in Hartshorne's AG): Let $Y\subseteq X$ be a subvariety. Let $\mathcal{O}_{Y,X}$ be the…
anon1234
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Why do we retain exactness when tensoring by $\mathcal{O}_C$ in Hartshorne, Lemma V.1.3?

Hartshorne, Algebraic Geometry, Chapter V, Lemma 1.3, reads (in part): Throughout this chapter, a surface will mean a nonsingular projective surface over an algebraically closed field $k$. [...] Let $X$ be a surface. [...] Lemma 1.3. Let $C$ be…
Daniel McLaury
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Example I.4.9.1 in Hartshorne (blowing-up)

Let $Y$ be the irreducible curve of $\mathbb{A}^2$ given by $y^2 = x^2(x+1)$. Let $t,u$ be homogeneous coordinates of $\mathbb{P}^1$. Then the total inverse image of $Y$ under the blowing-up $\phi: X \rightarrow \mathbb{A}^n$ of $\mathbb{A}^n$ at…
Manos
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Automorphism group of the configuration of lines on a cubic surface and quadratic transformations

It's well known that the automorphism group of the configuration of 27 lines on a smooth cubic surface in $\mathbb{P}^3$ (over a field containing all 27 lines) is isomorphic to the Coxeter group of type $ E_6 $. Labelling the vertices of $ E_6 $…
Will
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Total space of line bundle $\mathcal{O}(1)$ same as blow up of plane?

We recall the following facts about total spaces of bundles: Let $X$ be a scheme and $\mathcal{E}$ an invertible sheaf on $X$. The total space of $\mathcal{E}$, $\Bbb{V}(\mathcal{E})$ is defined as $\textbf{Spec} \operatorname{Sym}^\bullet…
user38268
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