Connection between complex curves and homology classes

[cduston]

Hey all,
This question stems from Scorpan, "The Wild World of 4-Manifolds", pg 302 (and all through that chapter). He states that a random homogeneous polynomial of degree d in CP^2 with coordinates $$[z_0:z_1:z_2]$$ defines a complex curve C, with homology class $$[C]=d[CP^1]$$.

So I understand that the homology classes of curves in CP^2 would be connected to the classes of CP^1, but I am a little surprised by the extra factor which is the degree of the curve. Is that simply because if you have some degree 2 homogeneous polynomial (say $$p(z_0,z_1,z_2)=z_0^2+z_1^2+z_2^2$$ ) then the homology class of CP^1 would have to "loop around twice" to cover the entire curve?

Any response would be appreciated.

 

[jvicens]

Thank you for your question. I am a mathematician specializing in algebraic geometry and I would be happy to provide some insight into your query.

Firstly, let me clarify that the statement made by Scorpan is referring to complex curves in complex projective space CP^2, not in the real space R^4. In CP^2, we have a notion of homology classes which are represented by closed curves, and these classes can be multiplied by integers to form new classes. This is what Scorpan is referring to when he says that the homology class of a curve C in CP^2 is equal to d times the homology class of CP^1.

To answer your question, yes, the extra factor of d in the homology class is due to the degree of the curve. This can be seen by considering the degree of a homogeneous polynomial, which is the highest power of its variables. In the case of a quadratic polynomial like the one you mentioned, the curve defined by the equation p(z_0,z_1,z_2) = 0 would indeed "loop around twice" in CP^2, and this is reflected in the homology class of the curve being equal to 2 times the homology class of CP^1.

In general, the degree of a curve in CP^2 is related to the number of times it intersects with a generic line in CP^2. This concept is known as the Bezout's theorem and it is a fundamental result in algebraic geometry. So, the degree of a curve not only affects its homology class, but it also tells us about the number of points of intersection with other curves in CP^2.

I hope this helps to clarify your understanding. If you have any further questions, please don't hesitate to ask. Thank you for your interest in this topic.