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Let $X$ be an algebraic variety and assume that there exists an $n\in\mathbb{N}$ and a closed immersion $\phi: X \hookrightarrow \mathbb{P}^n$. Now, is it a known and/or meaningful notion to define the degree of $X$ as the minimum of all degrees of such embeddings? In other words,

$\deg(X) := {\displaystyle \min_{ \substack{\phi: X \hookrightarrow \mathbb{P}^n \\\\ \text{closed immersion}}} \deg(\phi(X))}$

Another option I had in mind was taking $\deg(X)$ to be $\dim(X)!$ times the leading coefficient of the Hilbert Polynomial of its canonical ring. I would like to know if either notion is known, appears to have desirable properties and, possibly, if they are related.

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The Proj of the canonical ring might not be the same variety you started with. For example, with an ellitpic curve (or more generally a Calabi-Yau), the the proj of the canonical ring is a point. For varieties of general type, the Proj canonical ring will give you a variety of the same dimension and birational to the one you started with (at least in characteristic zero and if your given variety had nice enough singularities), but it will be a different variety in general. –  Karl Schwede May 2 '11 at 16:14
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Qiaochu, the canonical line bundle doesn't induce an embedding on a hyper-elliptic (genus $> 1$) curve, but a sufficently high power of it does. –  Karl Schwede May 2 '11 at 16:15
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@Qiaochu No. If $g>1$ then, for $n$ large, the number of sections of $n K$ is $n(2g-2) - g+1$, so the degree of the canonical ring is $2g-2$. Moreover, even in the case of a hyperelliptic curve, the canonical map is to a $\mathbb{P}^1$ of degree $g-1$ in $\mathbb{P}^{g-1}$, not to a degree $1$ projective line. –  David Speyer May 2 '11 at 16:27
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Your $X$ is going to have some ample cone, which may not be polyhedral but say it is. You can ask what the minimal faces are that have line bundles in them that are ample. It seems like there might be more than one such (though I don't have an example). Then those faces might not orthants, so there might not be a unique first interior point. I feel like picking one of the faces, then one of the interior points, based on the number deg($X$) is unlikely to have nice consequences. –  Allen Knutson May 2 '11 at 16:58
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A couple of obvious notes on the first definition, as also mentioned in comments to your previous question: It is certainly meaningful, and will necessarily be a positive integer. It will equal one iff $X$ is isomorphic to $\mathbb{P}^n$ for some $n$. Given that it is a fairly natural quantity to ask about, I'd be surprised if mathematicians have not considered it in the past; given the lack of responses here, I'm guessing that nothing particularly interesting was discovered about it. But I could easily be mistaken. –  Charles Staats May 2 '11 at 23:54
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