Genus Degree Formula for Curves over Arbitrary Fields and a Reference Request

algebraic-curvesalgebraic-geometryarithmetic-geometryfield-theoryreference-request

There exists the genus-degree formula for plane, projective, nonsingular curves that relates the (arithmetic) genus of a curve $C_F$ with the degree of the polynomial $F$ by the following relation:

$$\frac{1}{2} (\deg F – 1)(\deg F – 2) = g(C_F)$$

I know there are a lot of great resources and different ways to proof this. I found the statement in several algebraic geometry or curve theory books (Hartshorne, Milne, Fulton, etc.). My problem is that almost every resource that I have found so far has either only proofed this formula for algebraically closed fields or, if the field was arbitrary, was so scheme-theoretic that I wouldn't even recognise the formula if I saw it.

Two questions:

  1. Is this formula even true if the field is not closed (and/or not perfect)?
  2. If 1. is true, then what is a (citable) reference that states this formula for curves over arbitrary fields?

I will accept an answer that posts nothing but a citable source (maybe including the page) which contains the statement for arbitrary fields.

Best Answer

I wouldn't think that this would be likely to be written down in a citable fashion (too easy to make this an exercise). If I'm wrong, my saying so will surely increase the odds that someone will come along, correct me, and give you the answer you actually want. In the meantime, here is the proof: consider the long exact sequence on homology associated to $$0 \to \mathcal{O}_{\Bbb P^2_k}(-d) \to \mathcal{O}_{\Bbb P^2_k} \to \mathcal{O}_{C_F} \to 0.$$ By the calculation of the homology for sheaves of the form $\mathcal{O}_{\Bbb P^n_A}(d)$ for a ring $A$ (see Hartshorne theorem III.5.1, EGA III proposition 2.1.12, or Stacks 01XT) it is immediate to verify that $\dim_k H^0(\mathcal{O}_{C_F})=1$ and $\dim_k H^1(\mathcal{O}_{C_F})=\frac12(d-1)(d-2)$. Since the calculation of the cohomology of the sheaves $\mathcal{O}_{\Bbb P^n_A}(d)$ on $\Bbb P^n_A$ for a ring $A$ is independent of $A$, the result is true for any base field and for a bit broader definition of curve than you write (though you have left out probably the most important adjective here: you curve must be planar, i.e., a closed subscheme of $\Bbb P^2_k$).

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