Editing Moduli space (section)

===Projective space and Grassmannians===
The [[real projective space]] '''P'''<sup>''n''</sup> is a moduli space that parametrizes the space of lines in '''R'''<sup>''n''+1</sup> which pass through the origin. Similarly, [[complex projective space]] is the space of all complex lines in '''C'''<sup>''n''+1</sup> passing through the origin.

More generally, the [[Grassmannian]] '''G'''(''k'', ''V'') of a vector space ''V'' over a field ''F'' is the moduli space of all ''k''-dimensional linear subspaces of ''V''.

==== Projective space as moduli of very ample line bundles generated by global sections ====
Whenever there is an embedding of a scheme <math>X</math> into the universal projective space <math>\mathbf{P}^n_\mathbb{Z}</math>,<ref>{{Cite web|title=Lemma 27.13.1 (01NE)—The Stacks project|url=https://stacks.math.columbia.edu/tag/01NE|access-date=2020-09-12|website=stacks.math.columbia.edu}}</ref><ref>{{Cite web|title=algebraic geometry - What does projective space classify?|url=https://math.stackexchange.com/questions/296217/what-does-projective-space-classify|access-date=2020-09-12|website=Mathematics Stack Exchange}}</ref> the embedding is given by a line bundle <math>\mathcal{L} \to X</math> and <math>n+1</math> sections <math>s_0,\ldots,s_n\in\Gamma(X,\mathcal{L})</math> which all don't vanish at the same time. This means, given a point<blockquote><math>x:\text{Spec}(R) \to X</math></blockquote>there is an associated point<blockquote><math>\hat{x}:\text{Spec}(R) \to \mathbf{P}^n_\mathbb{Z}</math></blockquote>given by the compositions<blockquote><math>[s_0:\cdots:s_n]\circ x = [s_0(x):\cdots:s_n(x)] \in \mathbf{P}^n_\mathbb{Z}(R) </math></blockquote>Then, two line bundles with sections are equivalent<blockquote><math>(\mathcal{L},(s_0,\ldots,s_n))\sim (\mathcal{L}',(s_0',\ldots,s_n'))</math></blockquote>iff there is an isomorphism <math>\phi:\mathcal{L} \to \mathcal{L}'</math> such that <math>\phi(s_i) = s_i'</math>. This means the associated moduli functor <blockquote><math>\mathbf{P}^n_\mathbb{Z}:\text{Sch}\to \text{Sets}</math></blockquote>sends a scheme <math>X</math> to the set<blockquote><math>\mathbf{P}^n_\mathbb{Z}(X) =\left\{
(\mathcal{L},s_0,\ldots,s_n) : \begin{matrix}
\mathcal{L} \to X \text{ is a line bundle} \\
s_0,\ldots,s_n\in\Gamma(X,\mathcal{L}) \\
\text{ form a basis of global sections}
\end{matrix}
\right\} / \sim </math></blockquote>Showing this is true can be done by running through a series of tautologies: any projective embedding <math>i:X \to \mathbb{P}^n_\mathbb{Z}</math> gives the globally generated sheaf <math>i^*\mathcal{O}_{\mathbf{P}^n_\mathbb{Z}}(1)</math> with sections <math>i^*x_0,\ldots,i^*x_n</math>. Conversely, given an ample line bundle <math>\mathcal{L} \to X</math> globally generated by <math>n+1</math> sections gives an embedding as above.