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I am learning about degree of algebraic sets. I know the definition from Wikipedia https://en.wikipedia.org/wiki/Degree_of_an_algebraic_variety , but it is not too clear to me what it is. Could someone possibly explain to me what property exactly is degree of algebraic sets capturing (or how I should think about it)?

Thank you very much!

PS This was part of my other question Basic question regarding degrees of algebraic sets , but I thought maybe it would be better to ask this separately.

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Degree is best thought of as a property of projective varieties, since in $\mathbb{C}^n$, a linear space is not an intrinsic property. For example, in two variables, $x=0$ and $x=y^2$ define indistinguishable varieties, but the degrees of their equations are different. On the other hand, in $\mathbb{P}^n$ varieties are defined as zeroes of homogeneous polynomials. In particular, one would like to say that a hypersurface defined by a homogeneous polynomial of degree $d$ has degree $d$ as a variety. This can be translated as, if we intersect the hypersurface with a 'general' line, we should get $d$ points. So, the degree of an arbitrary projective variety of dimension $r$ in $\mathbb{P}^n$ is the number points in the intersection of the variety with a general linear subspace of dimension $n-r$. That this intersection is in fact a finite number of points and independent of the linear space as long as it is general are somewhat technical theorems. Of course, take some of this with a pinch of salt.

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  • $\begingroup$ Thank you for this helpful answer! Could you possibly explain a little more what is meant by general line? when you say: intersect the hypersurface with a 'general' line. Thanks! $\endgroup$ – Johnny T. Sep 15 '15 at 13:35
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    $\begingroup$ The set of lines in projective space is parametrized by a Grassmannian. So, this set has a natural variety structure. General means they come from a suitable non-empty open subset of this Grassmannian. Think of a quadric in projective plane. The special lines would be tangent lines, general lines the rest. $\endgroup$ – Mohan Sep 15 '15 at 14:01
  • $\begingroup$ Ok, sounds good. Thank you very much! By the way if you feel like it please look at my other question mentioned in the PS as well! :) It's also about degree of algebraic sets. I put a bounty but there hasn't been too much activity and it is going to expire soon... $\endgroup$ – Johnny T. Sep 16 '15 at 20:02

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