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Why we cannot define the degree of a quasi-projective $k$-variety ($k=\bar k$) $X$ for a given embedding $X\subset \mathbb P^n_k$ ?
If we take any compactification $\bar X$ of $X$, $\bar X\backslash X$ is a projective variety of dimension $\leq dim(X)-1$ so the intersection with a generic linear space of dimension $n-dim(X)$ is empty so why it seems not to be a standard fact?

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    $\begingroup$ Are you asking a question or proposing a definition? $\endgroup$
    – Jason Starr
    Jan 30, 2016 at 22:41
  • $\begingroup$ Good question, indeed... I am wondering why I do not see (so much) the definition of the degree for quasi-projective variety and explaining why I have some trouble understanding why there seems not to be such definition (because the natural one seems to work). $\endgroup$
    – user3001
    Jan 30, 2016 at 22:47

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I think this has a sociological, not mathematical, answer.

Whenever you have a structure that could be chosen on a mathematical object, it is handy to have one adjective to say you've made this choice (e.g. oriented) and another to say you could make this choice (e.g. orientable). It's really unfortunate that many, many such structures do not have two distinct adjectives to make clear whether the choice has been made.

Your definition is for a quasi-projectived variety (since you say "for a given embedding"), and is perfectly well-defined. In my experience the adjective quasiprojective is used to mean quasi-projectivable (i.e. there exists an embedding), not quasi-projectived. In which case the degree is not well-defined, because it depends on the embedding.

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  • $\begingroup$ Quasi-projectivized vs Quasi-projectived? $\endgroup$
    – AHusain
    Jan 31, 2016 at 3:37
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    $\begingroup$ I agree, and I would add two more remarks. (1) Of course algebraists and algebraic geometers do talk about the degree of quasi-projective varieties. For instance, the degree of an affine hypersurface, i.e., the total degree of the defining polynomial, is used constantly. (2) The degree of a projective variety is ubiquitous, e.g., in the Hilbert polynomial, in Bezout's theorem for plane curves, etc. For a quasi-projective variety some of these require care, yet others are not defined, e.g., the vector spaces of global sections of $\mathcal{O}(d)$ may be infinite-dimensional. $\endgroup$
    – Jason Starr
    Jan 31, 2016 at 8:05

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