What is the definition of a variety in Mumford's red book?

Question

In Mumford's red book, prevariety is defined (in II.3) as follows:

If $k$ is an algebraically closed field, a prevariety over $k$ is a reduced and irreducible prescheme of finite type over $k$.

Mumford then starts mentioning variety later on and I wasn't sure what it was exactly. I can not seem to find the definition in the book. Any clarification is appreciated. Thank you.

Answer 1

From the book, definition 2 in chapter I section 6 in the second edition (pg 37):

Definition 2 (Chapter I section 6, pg 37). Let $X$ be a prevariety. $X$ is a variety if for all prevarieties $Y$ and for all morphisms $$f,g:Y\rightrightarrows X$$ $\{y\in Y\mid f(y)=g(y)\}$ is a closed subset of $Y$.

The next proposition explains things a little:

Proposition 4 (Chapter I section 6, pg 37). Let $X$ be a prevariety. Then $X$ is a variety if and only if $\Delta(X)$ is closed in $X\times X$.

In chapter II, we have the same thing happening with schemes and preschemes:

Definition 2 (Chapter II section 6, pg 118): $f(x)\equiv g(x)$ if $f\circ i_x=g\circ i_x$, where $i_x:\operatorname{Spec} k(x)\to K$ is the canonical morphism. Equivalently, this means that $f(x)=g(x)$, and that the 2 maps $f^*_x,g^*_x:k(f(x))\to k(x)$ are equal.

Proposition 4 (Chapter II section 6, pg 118): For all $f,g:K\to X$, $$\{x\in k\mid f(x)\equiv g(x)\}$$ is locally closed.

Definition 3 (Chapter II section 6, pg 118): A prescheme $X$ is a scheme if for all preschemes $K$ and all $K$-valued points $f,g$ of $X$, $\{x\in K\mid f(x)\equiv g(x)\}$ is closed.

Proposition 5 (Chapter II section 6, pg 118): If $X$ is a prescheme over a ring $R$, then the criterion for $X$ to be a scheme is satisfied for all $K,f,g$ if it is satisfied in the case: $K=X\times_{\operatorname{Spec} R} X$, $f=p_1$, $g=p_2$.

Corollary 1 (Chapter II section 6, pg 119): If $k$ is an algebraically closed field, then a prevariety over $k$ is a variety in the sense of Ch. I if and only if it is a scheme.

In the modern language, this is the statement that $X$ is separated. So for Mumford, a variety is a separated prevariety. You'll find some other examples of this in older literature (use of pre-variety or pre-scheme to denote a possibly nonseparated variety or scheme), but this language has been out of fashion for many, many years.