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

Question

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.

Answer 1

The other answers are excellent and I found them very useful. I was going to accept one, but then I found a reference that contained exactly what I was looking for.

Qing Liu's Book Algebraic Geometry and Arithmetic curves in Chapter 7, Example 3.22

The formula is stated over arbitrary fields and explicitly.

Answer 2

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 $\Bbb P^2_k$).

Answer 3

Question: "Two questions: Is this formula even true if the field is not closed (and/or not perfect)? 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."

Answer: You also find a discussion of the arithmetic genus for a projective scheme of finite type over any field in Hartshorne, Ex.III.5.3. In ChI Ex.7.2, the arithmetic genus is defined using the Hilbert polynomial. In ChIII it is defined using the Euler characteristic $\chi(\mathcal{O}_X):=\sum_i h^i(X, \mathcal{O}_X)$ and the exercise proves that this equals the definition in ChI. Using the Hilbert polynomial it is not clear that the arithmetic genus is independent of choice of embedding, but the formula

$$p_a(X)=(-1)^n(\chi(\mathcal{O}_X)-1)$$

proves independence - the structure sheaf is an intrinsic invariant and does not depende on an embedding.

Note: The constructions of chapter I in Hartshorne are valid over an algebraically closed field $k$. The arithmetic genus $p_a(Y)$ is defined in Ex.I.7.2 for a projective variety $Y \subseteq \mathbb{P}^n_k$, hence $Y$ is by definition irreducible. By definition $p_a(Y):=(-1)^r(P_Y(0)-1)$ where $P_Y(t)$ is the Hilbert polynomial of the graded coordinate ring $S(Y)$ of $Y$ wrto the embedding $Y \subseteq \mathbb{P}^n_k$, hence with this definition it is not clear if $p_a(Y)$ is independent of choice of embedding: The coordinate ring $S(Y)$ is not an invariant of $Y$. In chapter III they give a general definition of the arithmetic genus $p_a(Y)$ for any closed subscheme $Y \subseteq \mathbb{P}^n_k$ over any field. Hence if you study a projective scheme over a non-algebraically closed field you must use the definition in III.Ex.5.3. For a curve $C \subseteq \mathbb{P}^2_k$ you get by definition

$$p_a(C):=1-h^0(C, \mathcal{O}_C)+h^1(C,\mathcal{O}_C),$$

and there are several methods to calculate this. You must somehow calculate $h^i(C,\mathcal{O}_C)$ and you find methods in the chapter on Cech cohomology. Here is a similar question with some suggestions:

Arithmetic genus of curves

Note (dec 2022) : This is also stated as an exercise in Hartshorne (HH) (Ex.III.4.7) in the chapter on cohomology. You must write down the Cech complex and calculate the dimensions explicitly. There is no condition on the base field or if the polynomial is irreducible. I do not think there is a "quick and easy method" as someone claims. The Hartshorne exercise reduces the case to covering the curve $C$ with two affine open subsets.

Here is a calculation using Ex.III.4.1 in HH: By Serre duality and Ex.III.4.1 in HH the following holds: Let $i:C \rightarrow S:=\mathbb{P}^2_k$ be the embedding into the projective plane. Since $i$ is an affine morphism it follows

$$H^i(C, \mathcal{O}_C) \cong H^i(S, i_*\mathcal{O}_C)$$

and

$$H^2(S, \mathcal{O}_S(-d)) \cong H^0(S, \mathcal{O}_S(d-3))^*$$

hence $h^2(S, \mathcal{O}_S(-d))=\frac{(d-1)(d-2)}{2}$ and $h^0(S, \mathcal{O}_S)=1$. Taking the long exact cohomology sequence of the sequence

$$ 0 \rightarrow \mathcal{O}_S(-d) \rightarrow \mathcal{O}_S \rightarrow i_*\mathcal{O}_C \rightarrow 0$$

you get the sequence

$$0 \rightarrow H^0(S, \mathcal{O}(-d)) \rightarrow H^0(S, \mathcal{O}_S) \rightarrow H^0(C, \mathcal{O}_C) \rightarrow $$

$$H^1(S, \mathcal{O}(-d)) \rightarrow H^1(S, \mathcal{O}_S) \rightarrow H^1(C, \mathcal{O}_C) \rightarrow $$

$$H^2(S, \mathcal{O}(-d)) \rightarrow H^2(S, \mathcal{O}_S) \rightarrow H^2(C, \mathcal{O}_C)=0.$$

It follows

$$H^0(C, \mathcal{O}_C) \cong H^0(S, \mathcal{O}_S) \cong k$$

and

$$H^1(C, \mathcal{O}_C) \cong H^2(S, \mathcal{O}(-d)) \cong H^0(S, \mathcal{O}(d-3))^*$$

hence

$$p_a(C)=1-h^0(C,\mathcal{O}_C)+h^1(C, \mathcal{O}_C)=$$

$$ 1-1+h^1(S,i_*\mathcal{O}_C) = h^2(S, \mathcal{O}(-d))-h^2(S, \mathcal{O}_S)=h^2(S, \mathcal{O}(-d))=h^0(S, \mathcal{O}(d-3)).$$

You get the formula

$$p_a(C)=\frac{(d-1)(d-2)}{2}.$$

Hence the result also follows from Ex.III.4.1 and III.5.1 in Hartshorne. In general a curve cannot be embedded into the projective plane, and for such curves the above does not apply.