Cartier divisor example in Harthsorne

Question

This question is about example 6.11.4 in Chapter II of Hartshorne. The example is about computing the Cartier divisor class group of the cuspidal cubic curve $y^2z = x^3$ in $\mathbb{P}_{k}^{2}$. He begins by saying "note that any Cartier divisor is linearly equivalent to one whose local function is invertible in some neighbourhood of the singular point $Z = (0,0,1)$".

I am not sure what this statement means at all. A Cartier divisor is, by definition, represented by a family $\{(U_{i}, f_{i}) \}$ where the $U_{i}$ cover $X$ and the $f_{i} \in \mathcal{K}^{*}(U_{i})$. But the elements of $\mathcal{K}^{*}(U_{i})$ are invertible by definition. So every local function is invertible, right? So what does Hartshorne mean by his statement?

Answer 1

I agree he did not phrase it very well. Let $U$ be an open chart containing the cusp. By removing the cuspidal point from the other charts, we may assume it is the only one containing the cuspidal point. Let $f$ be the rational function on $U$. What he means, is that we may assume that $f$ is invertible in $\mathcal{O}$ (and not in $\mathcal{K}$, which would be obvious as $\mathcal{K}$ is a field). So, he is assuming that $f \in \mathcal{O}^*(V)$, where $V \subset U$ is an open set containing the cusp.

The way to achieve it is the following. Notice that $f^{-1}$ is an element of $\mathcal{K}^*$, as so is $f$. Thus, $f^{-1}$ gives a principal Cartier divisor. So, the divisor $\lbrace (U_i,f_i) \rbrace$ is the same as $\lbrace (U_i,f_i \cdot f^{-1}) \rbrace$. Since $U = U_{i_0}$ for some index ${i_0}$, this shows that we may assume that on that chart the divisor is represented by $(U,1)$.

Answer 2

This is actually easier than it sound and I also agree with Stefano and you that it was not phrased very well. A generalization is also possible: You can choose up to linear equivalence Cartier divisor whose support (i.e. points where $f_i$ are not invertible) do not contain finite set of points. Shafarevich Basic Algebraic Geometry 1, p.153 has a proof of this in Chapter 3 under the subsection "Moving the Support of a Divisor away from a Point". I don't know how to cite this properly.. I would say Chapter 3, Section 1.3, Theorem 3.1. But I don't usually follow Shafarevich's proof as easily I do understand Hartshorne, like in this case it seems Shafarevich's proof is a bit confusing and goes through some local equations and I feel that in the proof one can avoid all these symbols.