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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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The Fano's plane and its homogenous coordinates

We let $k=\mathbb{Z}_{2}$. Is the assignment of the homogeneous coordinates $(0:0:1)$, $(0:1:0)$, $(1:0:0)$ to the main equilateral triangle of the Fano plane arbitrary? Could we for example start with the base of the triangle and ''name'' the…
H.E
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Hartshorne Exercise 2.6.

This might sound dumb to many people, but if I were to blindly try and carbon-copy the proof of Proposition 1.7 in Hartshorne to Exercise 2.6 where would I go wrong? Basically, I don't see where exactly that proof is failing? $\bf{Propositon 1.7}$…
V-B
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Vakil's exercise 5.5.#

I'd like to check that my solution to the following exercise of Vakil's FOAG is correct. I am bothered that we have to use (B) in it, and I want to make sure that I used it in the end for the correct reason. 5.5.E. EXERCISE (ASSUMING (A) AND (B)).…
Rodrigo
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Example of a function on a locally noetherian $X$, one of the irreducible components of whose support is the union of more than one associated point

Let $X$ be a locally noetherian scheme. Ravi Vakil points out on page 167 of FOAG that Each of the irreducible components of the support of any function on a locally Noetherian scheme is the union of the closures of some subset of the …
Rodrigo
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Finding a disjoint subvariety

Given sub-variety $X\subset \mathbb{A}^{2k}$ of dimension $k-1$, how can I find a sub-variety $Y\subset \mathbb{A}^{2k}$ of the same dimension which is disjoint to $X$? Perhaps I should mention I read Kempf's 'algebraic varieties' up to chapter 5.
pumpam
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Ext -vanishing for sheaves

For $X$ a smooth, projective variety one has that for two coherent sheaves $\mathcal{F}$ and $\mathcal{G}$, $\mathrm{Ext}^i(\mathcal{F},\mathcal{G})=0$, for $i>>0$. Do we really need projectivity of $X$ or does this hold also for quasi-projective…
user109227
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Why cubic surfaces contain straight lines?

I often heard that each smooth cubic surface contains even $27$ straight lines. I cannot prove it today, but I'll do my best to do it soon. However how to prove that each cubic surface contains a straight line?
user74574
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The two possible structures on a triple point

Let $k$ be an algebraically closed field. I would like to prove that there are only two possible $k$-scheme structures on a triple point, namely that of a $2$-jet $\textrm{Spec }k[x]/(x^3)$, and that of $\textrm{Spec }k[x,y]/(x^2,xy,y^2)$. A triple…
Brenin
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Scheme theoretic image of a base change of a morphism of schemes

Let $f\colon X \rightarrow Y$, $g\colon Y' \rightarrow Y$, be two morphisms of schemes. Let $X' = X\times_Y Y'$, and let $f'\colon X' \rightarrow Y'$ be the projection. We are interested in the relation between the scheme theoretic image of $f'$ and…
quim
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Understanding the Affine Case of a Stacky Result

I'm going through Vistoli's sections of FGA Explained to begin to understand stacks. It is well-known and proven in the text that the fibered category $QCoh$ of quasi-coherent sheaves is a stack in the fpqc topology. In particular then, given two…
Cass
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Generalizations of Hilbert's Syzygy theorem

Hilbert's Syzygy theorem states that a minimal free resolution of a finitely generated graded module over a (standard graded) polynomial ring in $n$ variables $k[x_1, \ldots, x_n]$ does not have more than $n+1$ terms in it. To what rings other than…
Carl
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A Question on "flatness is preserved by base change"

Let $f:X\to Y$ be a morphism, $\mathcal{F}$ be an $O_X$ module which is flat over $Y$, let $g:Y'\to Y$ be any morphism. Let $X'=X\times_YY'$, let $f':X'\to Y'$ the second projection, and $\mathcal{F'}=p_1^*(\mathcal{F})$ .Then $\mathcal{F'}$ is flat…
user93417
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Some questions on Hartshorne III Ex 6.8

I have been looking at Hartshorne III exercise 6.8 for nearly a week now and I don't seem to have a clue as to how to do it. In particular, I am stuck on part (a) which boils down to showing the following. Let $X$ be integral, separated and…
user38268
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Relation of Between inseparable morphisms and tangent lines

$X$ be a projective curve of $\mathbb{P}^n$, $P$ is not $X$ and let $f:X\rightarrow \mathbb{P}^{n-1}$ be the projection from $P$. When I read Hartshorne book, I see that if $f$ is insepable (i.e the function field of $X$ is inseparable extension of…
Sang Cheol Lee
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How to define the union of closed subschemes in an affine scheme

How to define the union of closed subschemes in an affine scheme? Suppose $I$ and $J$ define closed subschemes of $\operatorname{Spec}R$, how should we define their intersection? Eisenbud and Harris (GTM 197, p24) defined it by $I\cap J$ and used…
user93417
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