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Edit (May 12, 2022): While it is probably too late for the purpose of the original posting, but I also wanted a more "direct" way (compared to the approach by Hilbert polynomial) to see degree as the number of points in the intersection with generic linear subspaces with complementary dimension. I could not find a reference, and eventually ended up writing it up myself (see Example III.122 and Theorem III.123 in Mondal - How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft III)). It does not use Bertini's theorem, so in positive characteristic needs to count points of intersection with appropriate multiplicities. I am happy to report that while I had forgotten about this question, I actually ended up doing precisely what I suggested 10 years ago, namely following Mumford's proof from "Algebraic Geometry I".

Hi Kiumars, this does not answer your question, since the proof works only when the field is $\mathbb{C}$, but when I was learning it, the most accessible and understandable proof for this case was that of Theorem 5.1 of Mumford's "Algebraic Geometry I: Complex Projective Varieties".

Edit (May 12, 2022): While it is probably too late for the purpose of the original posting, but I also wanted a more "direct" way (compared to the approach by Hilbert polynomial) to see degree as the number of points in the intersection with generic linear subspaces with complementary dimension. I could not find a reference, and eventually ended up writing it up myself (see Example III.122 and Theorem III.123 in Mondal - How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft III)). It does not use Bertini's theorem, so in positive characteristic needs to count points of intersection with appropriate multiplicities. I am happy to report that while I had forgotten about this question, I actually ended up doing precisely what I suggested 10 years ago, namely following Mumford's proof from "Algebraic Geometry I".

This does not answer your question, since the proof works only when the field is $\mathbb{C}$, but when I was learning it, the most accessible and understandable proof for this case was that of Theorem 5.1 of Mumford's "Algebraic Geometry I: Complex Projective Varieties".

Edit (May 12, 2022): While it is probably too late for the purpose of the original posting, but I also wanted a more "direct" way (compared to the approach by Hilbert polynomial) to see degree as the number of points in the intersection with generic linear subspaces with complementary dimension. I could not find a reference, and eventually ended up writing it up myself (see Example III.122 and Theorem III.123 in arxiv:1806.05346). It does not use Bertini's theorem, so in positive characteristic needs to count points of intersection with appropriate multiplicities. I am happy to report that while I had forgotten about this question, I actually ended up doing precisely what I suggested 10 years ago, namely following Mumford's proof from "Algebraic Geometry I".

Hi Kiumars, this does not answer your question, since the proof works only when the field is $\mathbb{C}$, but when I was learning it, the most accessible and understandable proof for this case was that of Theorem 5.1 of Mumford's "Algebraic Geometry I: Complex Projective Varieties".

Edit (May 12, 2022): While it is probably too late for the purpose of the original posting, but I also wanted a more "direct" way (compared to the approach by Hilbert polynomial) to see degree as the number of points in the intersection with generic linear subspaces with complementary dimension. I could not find a reference, and eventually ended up writing it up myself (see Example III.122 and Theorem III.123 in Mondal - How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft III)). It does not use Bertini's theorem, so in positive characteristic needs to count points of intersection with appropriate multiplicities. I am happy to report that while I had forgotten about this question, I actually ended up doing precisely what I suggested 10 years ago, namely following Mumford's proof from "Algebraic Geometry I".

Hi Kiumars, this does not answer your question, since the proof works only when the field is $\mathbb{C}$, but when I was learning it, the most accessible and understandable proof for this case was that of Theorem 5.1 of Mumford's "Algebraic Geometry I: Complex Projective Varieties".

Hi Kiumars, this does not answer your question, since the proof works only when the field is $\mathbb{C}$, but when I was learning it, the most accessible and understandable proof for this case was that of Theorem 5.1 of Mumford's "Algebraic Geometry I: Complex Projective Varieties".

Edit (May 12, 2022): While it is probably too late for the purpose of the original posting, but I also wanted a more "direct" way (compared to the approach by Hilbert polynomial) to see degree as the number of points in the intersection with generic linear subspaces with complementary dimension. I could not find a reference, and eventually ended up writing it up myself (see Example III.122 and Theorem III.123 in arxiv:1806.05346). It does not use Bertini's theorem, so in positive characteristic needs to count points of intersection with appropriate multiplicities. I am happy to report that while I had forgotten about this question, I actually ended up doing precisely what I suggested 10 years ago, namely following Mumford's proof from "Algebraic Geometry I".

Hi Kiumars, this does not answer your question, since the proof works only when the field is $\mathbb{C}$, but when I was learning it, the most accessible and understandable proof for this case was that of Theorem 5.1 of Mumford's "Algebraic Geometry I: Complex Projective Varieties".