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.

29074 questions
5
votes
1 answer

Injective $\mathcal{O}_X$ module is flasque

This is a lemma from Hartshorne. Let $(X,\mathcal{O}_X)$ be a ringed space. Then any injective $\mathcal{O}_X$ module is flasque. I am trying to prove this. Let $V\subseteq U$ be open subsets of $X$, we need to prove that the restriction map…
user312648
5
votes
1 answer

Coherent sheaves on a non-singular algebraic variety

Grothendieck wrote in his letter to Serre(Nov. 12,1957) that every coherent algebraic sheaf on a non-singular algebraic variety(not necessarily quasi-projective) is a quotient of a direct sum of sheaves defined by divisors. I think "sheaves defined…
Makoto Kato
  • 42,602
5
votes
1 answer

Is a non-singular connected algebraic variety irreducible?

I mean by an algebraic variety a locally closed(in Zarisky topolgy) subset of a projective space over an algebraically closed field. If this is not the case, I would like to know counter-examples in dimension both one and two over the field of…
Makoto Kato
  • 42,602
5
votes
2 answers

Finding the regular points of a rational map

Let $X$ and $Y$ be (irreducible, quasi-projective) varieties (over an algebraically closed field $k$), and $\phi: X \to Y$ be a rational map. I think I understand what it means for $\phi$ to be regular at a point $x \in X$, but I'm lost as to how to…
Zhen Lin
  • 90,111
5
votes
1 answer

Conormal Sheaf (Morphisms of Schemes, Stacks Project)

The first part of this question refers to Lemma 33.2 from the chapter "Morphisms of Schemes" of the Stacks Project. In particular, if $i: Z \rightarrow X$ is an immersion and $\mathcal{I}$ is the corresponding ideal sheaf, then the conormal sheaf is…
Manos
  • 25,833
5
votes
1 answer

Is this subvariety affine?

Consider the following algebraic variety $X$: Let $C\subset\mathbb{A}^3$ be the affine variety cut out by $xy-z^2=0$ (i.e. the affine cone over a projective conic). Let $B=\mathbb{A}^2$, with coordinate functions $x',z'$. Identify $C\setminus…
Ben Blum-Smith
  • 20,766
5
votes
1 answer

Help with irreducible components

I want to find the irreducible components of the algebraic set $Y$, in $\mathbb{A}^{3}$ given by the zero-locus of the equations $x^{2}-yz$ and $xz-z$. I also want to compute the dimension of $Y$. Well if $xz-z=0$ then $x=1$ or $z=0$. If $z=0$ then…
user6495
  • 3,957
5
votes
1 answer

Criterion for a subvariety to be affine?

Say $X$ is an algebraic variety and $U\subset X$ is open. Consider the natural map $U\rightarrow \operatorname{Spm}(\mathcal{O}_X(U))$ given by sending a point of $U$ to the ideal of sections over $U$ that vanish on that point. $U$ is affine if…
Ben Blum-Smith
  • 20,766
5
votes
1 answer

Local rings are not 'really local' in Zariski topology

My professor made a comment the "Local rings are not really local in the Zariski topology (on say $\mathbb{C}^{n}$). Thus we take completions of local rings at all points of a smooth variety which turn out to be the same. However in the classical…
Shravan Patankar
  • 161
5
votes
2 answers

Quasi-coherent ideals and subschemes

Let $X$ be a scheme. Let $\mathcal{I}$ be a quasi-coherent ideal of $\mathcal{O}_X$. Let $Y = Supp(\mathcal{O}_X/\mathcal{I})$. Let $f\colon Y \rightarrow X$ be the canonical injection. Then how do we prove that $(Y,…
Makoto Kato
  • 42,602
5
votes
0 answers

Connected variety with a $k$-rational point is geometrically connected

Let $X$ be a connected scheme of finite type over a field $k$. I'm trying to understand the following three statements: (i): If $X$ has a $k$-rational point, then $X_{k'} = X \times_k k'$ is connected for any finite extension of $k$. (ii): The same…
D_S
  • 33,891
5
votes
2 answers

Schemes as set valued functors on the category of affine schemes

Let $\mathcal{Sch}$ be the category of schemes. Let $\mathcal{Aff}$ be the category of affine schemes. Let $\mathcal{Sets}$ be the category of sets. Let $X$ be a scheme. Let $h_X\colon \mathcal{Sch}^{op} \rightarrow \mathcal{Sets}$ be the functor…
Makoto Kato
  • 42,602
5
votes
0 answers

Principal divisor on a projective curve has degree zero

In the book "Algebraic Geometry" of Hartshorne, in corollary 6.10 chapter 2, he wants to prove that principal divisors have degree zero. To do this he takes $f\in K(X)^*$ and this gives a morphism $\phi: X\rightarrow \mathbb{P}^1$. Now he says that…
SC30
  • 576
5
votes
1 answer

Global sections of a sheaf restricted to a closed subscheme

Let $X$ be non-singular projective variety. Consider a smooth closed subvariety say $Y$ of $X$. Let $F$ be a torsion-free coherent sheaf on $X$. Then is there any description of global sections of $F|_Y$ in terms of global sections of $F$ on…
user349424
  • 738
5
votes
1 answer

Points of bounded degree on varieties

Let $\mathbb{Q}^{alg}$ be the algebraic closure of the rationals. Given a point $P\in \mathbb{A}^n(\mathbb{Q}^{alg})$, $P = (a_1,\dots,a_n)$, we define the degree of $P$ to be the degree of the minimal field extension of $\mathbb{Q}$ over which $P$…
Alex Kruckman
  • 76,357
1 2 3
…
99 100