## 43.2 Conventions

We fix an algebraically closed ground field $\mathbf{C}$ of any characteristic. All schemes and varieties are over $\mathbf{C}$ and all morphisms are over $\mathbf{C}$. A variety $X$ is nonsingular if $X$ is a regular scheme (see Properties, Definition 28.9.1). In our case this means that the morphism $X \to \mathop{\mathrm{Spec}}(\mathbf{C})$ is smooth (see Varieties, Lemma 33.12.6).

Comment #4864 by Matthieu Romagny on

I find the choice of the letter $\mathbb{C}$ for the algebraically closed field quite confusing. I happened to be reading parts of this chapter without prior knowledge of the conventions section, and it really puzzled me. Why not just $k$?

Comment #5150 by on

@#4864: Yes, this wasn't a good choice. The idea was that we should not use $k$ as this is usually a random (not algebraically closed) field.

Related comment: with relatively little work we can extend almost all the discussion in this chapter to nonalgebraically closed fields. The only tricky thing is to extend the moving lemma to the case where the ground field is finite where you have to do a bit of work. Anway, if we do this, then we can use $k$.

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