# Qing Liu's definition of an algebraic variety, a non-separated line

First, a little reminder. In Qing Liu's Book on algebraic Curves, algebraic varieties are defined as

Let $k$ be a field. An affine variety over $k$ is the affine scheme associated to a finitely generated algebra over $k$. An algebraic variety over $k$ is a $k$-scheme $X$ such that there exists a covering by a finite number of affine open subschemes $X_i$ which are affine varieties over $k$. A projective variety over $k$ is a projective scheme over $k$.

a little later, he describes the well-known example (Example 3.3.8) of the line with a doubled point at the origin (see also Hartshorne, II.2.3.6):

$$-------:-------$$

Now as I understand it, this line is not separated and should therefore not be an algebraic variety. I just don't see where Liu's definition fails this example to be an algebraic variety.

• It is a variety by Liu's definition. If you want varieties to be separated, you have to impose that condition somewhere! – user64687 Feb 27 '15 at 10:47
• So I might add the separability in my texts. Compared to the definition Hartshorne gives, this is still pretty weak. – Dan Feb 27 '15 at 14:41
• This is a big problem with the notion of "variety" that has bugged me for quite some time as well: There is no really good consensus about what the word precisely means. – Jesko Hüttenhain Feb 27 '15 at 14:50
• Do you mean Hartshorne's definition on Page 105? Yes, that is quite strong compared to Liu's. Each definition has its advantages and disadvantages: Liu does not appear to require varieties to be reduced, which feels a bit weird to me; on the other hand, Hartshorne's definition requires them to be irreducible, which means that for example the limit of a family of varieties might not be a variety. – user64687 Feb 27 '15 at 14:52
• I think the main thing to keep in mind is that authors make their definitions in the context of what they want to achieve. Hartshorne's definition is meant to reflect the "classical" point of view on algebraic geometry, and is appropriate for his Chapters 4 and 5; Liu's book is obviously oriented more towards arithmetic, and his definition reflects that --- e.g. not requiring $k$ to be closed. – user64687 Feb 27 '15 at 14:55