most recent 30 from http://mathoverflow.net 2013-05-23T11:56:01Z http://mathoverflow.net/feeds/question/103657 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103657/degree-of-a-variety-is-well-defined Kiumars 2012-08-01T04:53:43Z 2012-08-01T22:31:23Z

<p>Let $X$ be a projective variety embedded in a projective space, over a field of arbitrary characteristic. </p> <p>What is a good reference for a nice proof of the classic fact that degree of $X$ is well-defined, that is, a "general" plane of complementary dimension intersects $X$ in a fixed number of points? (assuming Bertini's theorem). A proof which requires less amount of background is preferable.</p>

http://mathoverflow.net/questions/103657/degree-of-a-variety-is-well-defined/103673#103673 Damian Rössler 2012-08-01T09:08:36Z 2012-08-01T09:08:36Z

<p>See Fulton, "Intersection Theory", Ex. 8.4.12, p. 149.</p>

http://mathoverflow.net/questions/103657/degree-of-a-variety-is-well-defined/103675#103675 auniket 2012-08-01T09:45:02Z 2012-08-01T09:45:02Z

<p>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".</p>

http://mathoverflow.net/questions/103657/degree-of-a-variety-is-well-defined/103678#103678 Kaveh 2012-08-01T11:43:35Z 2012-08-01T11:43:35Z

<p>I suppose one of the best approaches (In my humble view, the best one) to delve into the concept of the degree of a projective variety is the method that has been used in "Algebraic Geometry" by Robin Hartshorne. </p> <p>More precisely, if you study Chapter 1, Section 7 (Intersections in Projective Space), it will give you a real insight into the matter. According to the fact that, as it is mentioned in the aforementioned part of the book, a purely algebraic definition of degree has been given, it would be much easier(at least to me) to gain a clear answer to your question. </p> <p>I would like to mention that Theorem 7.5 ( Hilbert- Serre) plays a vital role in the aforesaid section and if you are already familiar with the Hilbert polynomial of a graded module over a polynomial ring and its Hilbert function, you can directly go to Hilbert- Serre Theorem and the rest of the section. Otherwise have a quick look at the whole section and in a few minutes you will get what you want.</p> <p>I hope it will help.</p>

http://mathoverflow.net/questions/103657/degree-of-a-variety-is-well-defined/103735#103735 Sándor Kovács 2012-08-01T22:31:23Z 2012-08-01T22:31:23Z

<p>As Kaveh says, the best way is to follow Hartshorne's Chapter I, Section 7. This is before he gets to schemes, so it should be considered "classical".</p> <p>Thm 7.7 in that section is a slightly generalized version of Bezout's (classical) theorem on the number of points plane curves intersect in. It considers intersections with hypersurfaces. Now let $X\subseteq \mathbb P^n$ and its degree defined via its Hilbert polynomial (as on page 52). Then by (the proof of) Bertini's theorem, for a general hyperplane $H\subseteq \mathbb P^n$, $X\cap H$ is reduced and then by Thm. 7.7, $\deg (X\cap H)=\deg X$. Keep cutting with hyperplanes until you get to finitely many points and it is easy to check that the degree of a reduced zero-dimensional scheme is the same as the number of points in it.</p>