# Definition: The Germ of a Variety at a Point

algebraic-geometryalgebraic-topology

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!

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 $$\text{Spec } \mathcal O_{Z,p}$$