This should be a simple thing to find by pure googling, and yet, I keep finding nothing that is actually helpful.

I'm having trouble understanding the notion of *the* germ of a variety at a point.

I certainly understand what *a* germ of a function at a given point on a variety is, but in the paper that I'm trying to decipher, Gonzalez-Sprinberg and Verdier's famous * Construction géométrique de la correspondance de McKay*, the germ in question certainly appears to be a geometric object, as opposed to a function.

What little I found suggested that *the germ of a variety at a point* was, in some sense, an equivalence class of varieties in the same ambient space that locally look similar around the point in question. However, since the paper *immediately* goes on to talk about the Picard group of the germ, I concluded that this cannot be the case, because two varieties may locally look the same around a given point, but have different Picard groups. Thus, such a notion would be ill-defined.

As always, I look forward to your responses!

## Best Answer

Let $X$ be your variety. A germ of a variety at $p$ is given by an open subscheme $U$ containing $p$ and a closed subscheme of $U$, $Z$. $Z$ defines an ideal sheaf in $\mathcal O_U$, $\mathcal I_{Z}$ and therefore one can speak about the stalk of $Z$ at $p$, $\mathcal I_{Z,p} \subset \mathcal O_{U,p} = \mathcal O_{X,p}$.

By the very definition of the stalk at a point, you can see that two germs $Z$, $Z'$ are equivalent if and only if $\mathcal I_{Z,p}=\mathcal I_{Z',p}$. If $U=\text{Spec }A$ is an affine neighbourhood of $p$, by commutative algebra every ideal of $A_p = \mathcal O_{X,p}$ comes form an ideal in $A$, which defines a germ of a variety at $p$. On the other hand, you know that ideals of a ring are in $1-1$ correspondence with closed subschemes of its specrum.

Thus, we have that

$$\left\lbrace \text{germs of varieties at } p \text{ up to equivalence of germs} \right\rbrace \leftrightarrow \left\lbrace \text{closed subschemes of Spec }\mathcal O_{X,p} \right\rbrace $$

In that snese, the germ of a variety defines a unique closed subscheme, but not in $X$ but in another variety, $\text{Spec } \mathcal O_{X,p}$.

The Picard group that the text refers to is the Picard group of this closed subscheme. Now, I leave to you to check that the closed subscheme of $\text{Spec } \mathcal O_{X,p}$ corresponding to the germ $Z$ is isomorphic to $\text{Spec } \mathcal O_{Z,p}$