2
$\begingroup$

In the example 1.24 of the book "Projective Duality and Homogeneous Spaces"- A.E. Tevelev the author says this:

Let us recall the construction of invertible sheaves on projective spaces P(V). All these sheaves have the form $O(d)$, where $O(d)$ is the sheaf of homogeneous functions of degree $d$ on $P(V)$. More precisely, let $n : V\setminus\lbrace 0 \rbrace \longrightarrow P(V)$ be the canonical projection. If $U \subseteq P(V)$ is a Zariski open set, then the sections of $O(d)$ over $U$ are, by definition, regular functions $f$ on $\pi^{-1}(U) \subseteq V$, which are homogeneous of degree $d$.

Mi question is the following:

In general given a homogeneous function $f$, it's value on an eq. class of $P(V)$ is not well defined, then... how a homogenous function on $\pi^{-1}(U)$ defines a function on $U$?

  • 3
    They don't define functions on $U$. The point is that the expression "section of $O(d)$ on U" means "homogeneous function of degree $d$ on $\pi^{-1}(U)$", the latter of which has well-defined values. While of course the values don't descend to $U$, the zero loci do, which are what we typically work with. – 2017-01-11
  • 1
    Tabes's interpretation is correct. Recall that the data of a sheaf $\mathcal{F}$ is a collection of groups of sections $\mathcal{F}(U)$ for all open subsets $U$ which satisfy restriction and gluing axioms. In the paragraph these groups are defined for the sheaf $\mathcal{O}(d)$. All this of course is very standard. One thing to keep in mind is that $\mathcal{O}(-1)$ is the sheaf of sections of the tautological line bundle over $\mathbb{P}(V)$. – 2017-01-15

0 Answers 0