As we know, cohomology of sheaves on affine schemes is boring. The next simplest case we can consider is sheaves on projective schemes.
7. The \( \operatorname{Proj} \) construction#
We can define projective space by gluing affine space together, but there is a more canonical construction. By a graded ring \( S \), we implicitly assume that \( S = \bigoplus _ {d \ge 0} S _ d \).
Definition 1. The scheme \( \operatorname{Proj} S \) is defined as the following. A point in \( \operatorname{Proj} S \) is a homogeneous prime ideal \( \mathfrak{p} \) of \( S \) such that \( \mathfrak{p} \) doesn’t contain all of \( S _ 1, S _ 2, \ldots \). The topology is generated by the sets \( D(f) = \{ \mathfrak{p} : f \notin \mathfrak{p}\} \) for homogeneous \( f \in S _ d \) with \( d \ge 1 \), and the structure sheaf on \( D(f) \) is given by the \( \mathrm{Spec} \) of the \( 0 \)-th graded piece \( (S _ f) _ 0 \).
Example 1. If \( S = A[x _ 0, \ldots, x _ n] \), we have \( \operatorname{Proj} S = \mathbb{P} _ A^n \).
So any kind of section needs to come from a \( 0 \)-th graded element in the ring. This makes sense, because on projective space, functions like \( x \) or \( y^2 / (z+1) \) don’t give well-defined values. On affine schemes, we were able to construct sheaves on \( \mathrm{Spec} A \) from \( A \)-modules. Similarly, we can construct sheaves on \( \operatorname{Proj} S \) from graded \( S \)-modules.
Definition 2. Let \( S \) be a graded ring, and let \( M = \bigoplus _ n M _ n \) be a graded \( S \)-module. We define a quasi-coherent sheaf \( \tilde{M} \) on \( \operatorname{Proj} S \), so that \( \tilde{M} \vert _ {D(f)} \cong ((M _ f) _ 0)^{\tilde{}} \).
The sheaves are glued via, where \( \deg f = a \) and \( \deg g = b \), the isomorphisms \( ((M _ f) _ 0) _ {g^a/f^b} \cong (M _ {fg}) _ 0 \cong ((M _ g) _ 0) _ {f^b / g^a} \). We immediately see that this is a quasi-coherent sheaf on \( \operatorname{Proj} S \). Of course, this construction is functorial in \( M \), and a short exact sequence \( 0 \rightarrow M _ 1 \rightarrow M _ 2 \rightarrow M _ 3 \rightarrow 0 \) induces an exact sequence \[ \displaystyle 0 \rightarrow \tilde{M} _ 1 \rightarrow \tilde{M} _ 2 \rightarrow \tilde{M} _ 3 \rightarrow 0 \] of quasi-coherent sheaves, because the construction is exact affine-locally.
But for graded modules, we have an interesting operation called "twisting". Namely, for each integer \( n \), we can shift the indices to define a new graded module \( M(n) \), so that \[ \displaystyle M(n) _ k = M _ {n+k}. \] This makes huge difference, because sections are always supposed to be degree \( 0 \) elements, and are now degree \( n \) elements of \( M \). An analogous definition can be made for quasi-coherent sheaves in general.
Definition 3. Let \( \mathscr{F} \) be a quasi-coherent sheaf on \( X = \operatorname{Proj} S \), where \( S \) is a graded ring generated by \( S _ 1 \) as an \( S _ 0 \)-algebra. Note that this condition ensures that \( X = \bigcup _ {f \in S _ 1} D(f) \). For each \( f \) homogeneous of degree \( 1 \), we let \( \mathscr{F}(n) _ f = \mathscr{F}(n) \vert _ {D(f)} \cong \mathscr{F} \vert _ {D(f)} \). But we glue them by the isomorphisms \( \mathscr{F}(n) _ f \vert _ {D(fg)} \rightarrow \mathscr{F}(n) _ g \vert _ {D(fg)} \) given by \( s \mapsto s \cdot f^n / g^n \). On triple intersections, the gluing maps commute, so this indeed defines a quasi-coherent sheaf.
Let \( S \) be a graded ring generated by \( S _ 1 \) as an \( S _ 0 \)-algebra, and let \( X = \operatorname{Proj} S \). Here are some properties that can be easily verified: (Hartshorne [Har77] III.5.12)
- For any graded \( S \)-module \( M \), we have \( \tilde{M}(n) \cong (M(n))^{\tilde{}} \).
- The sheaf \( \mathscr{O} _ X(n) \) is invertible, i.e. locally free, for all \( n \).
- We have \( \mathscr{O} _ X(n) \otimes _ {\mathscr{O} _ X} \mathscr{F} \cong \mathscr{F}(n) \) for every quasi-coherent sheaf \( \mathscr{F} \).
One important remark is that to define \( \mathscr{F}(n) \), we need to specify what \( S \) is. If \( \operatorname{Proj} S \cong \operatorname{Proj} T \) for two graded rings \( S \) and \( T \), the sheaves \( \mathscr{F}(n) \) can be defined with either of \( S \) and \( T \), and there is no reason for them to be isomorphic.
Over a general scheme, we can also define all these twisted sheaves \( \mathscr{F}(n) \), as long as we have chosen an invertible sheaf \( \mathscr{O} _ X(1) \), because we can define \( \mathscr{F}(n) = \mathscr{F} \otimes \mathscr{O} _ X(1)^{\otimes n} \). If \( X \) is defined over \( Y \), an invertible sheaf \( \mathscr{L} \) is called very ample if there is a locally closed immersion \( i : X \rightarrow \mathbb{P} _ Y^n \) such that \( \mathscr{L} = i^\ast(\mathscr{O}(1)) \). We can develop the theory of twisted sheaves in this setting, but I prefer not to.
Example 2. Let us take the projective space \( S = A[x _ 0, \ldots, x _ n] \) and \( X = \mathbb{P}^n = \operatorname{Proj} S \), and try to compute cohomology of \( \mathcal{O} _ X(m) \). On each open \( D(x _ i) \cong \mathrm{Spec} A[x _ 0 / x _ i, \ldots, x _ n / x _ i] \), the sheaf \( \mathscr{O} _ X(m) \) looks like the homogeneous degree \( m \) polynomials in \( A[x _ 0, \ldots, x _ n, x _ i^{-1}] \). Then the Čech complex looks like the degree \( m \) components of \[ \displaystyle 0 \rightarrow \prod _ {i}^{} A[x _ 0, \ldots, x _ n, x _ i^{-1}] \rightarrow \prod _ {i < j}^{} A[x _ 0, \ldots, x _ n, x _ i^{-1}, x _ j^{-1}] \rightarrow \cdots. \] Because each is a direct sum of a copy of \( A \) corresponding to each (Laurent) monomial, we can compute them separately and arrive at \( H^k(\mathbb{P} _ A^n, \mathscr{O} _ X(\bullet)) = 0 \) for \( 0 < k < n \) and \[ \displaystyle \begin{aligned} H^0(\mathbb{P} _ A^n, \mathscr{O} _ X(\bullet)) & \displaystyle\cong A[x _ 0, \ldots, x _ n], \\ H^n(\mathbb{P} _ A^n, \mathscr{O} _ X(\bullet)) & \displaystyle\cong (x _ 0 \cdots x _ n)^{-1} A[x _ 0^{-1}, \ldots, x _ n^{-1}]. \end{aligned} \] So \( H^0(\mathbb{P} _ A^n, \mathscr{O} _ X(m)) \) is a free \( A \)-module of rank \( \binom{m+n}{n} \) for \( m \ge 0 \) and \( H^n(\mathbb{P} _ A^n, \mathscr{O} _ X(m)) \) is free of rank \( \binom{-m-1}{n} \) for \( m \le -n-1 \).
8. Between graded modules and quasi-coherent sheaves#
We have seen how to construct a quasi-coherent sheaf from a module. In the affine case, there was an equivalence of categories between modules and quasi-coherent sheaves. To go from quasi-coherent sheaves to modules, we only needed to take the global sections. Will something similar happen over \( \operatorname{Proj} S \) as well?
Definition 4. Let \( \mathscr{F} \) be a quasi-coherent sheaf over \( X = \operatorname{Proj} S \), where \( S \) is generated by \( S _ 1 \) as an \( S _ 0 \)-algebra. Note that any element \( f \in S _ n \) can be thought of as a global section in \( \mathscr{O} _ X(n) \). Then we define a graded \( S \)-module \( \Gamma(\mathscr{F}) \) by \[ \displaystyle \Gamma(\mathscr{F}) _ m = \Gamma(X, \mathscr{F}(m)). \] The multiplication structure is given by \( (f, s) \mapsto f \otimes s \in \Gamma(X, \mathscr{O} _ X(m) \otimes \mathscr{F}(n)) = \Gamma(\mathscr{F}) _ {m+n} \).
In general, if \( s \in M _ m \) is any homogeneous element, it can be regarded as a global section in \( \Gamma(X, \mathscr{F}(m)) \) because it can be viewed locally and can be glued well. Conversely, if \( \mathscr{F} \) is a quasi-coherent sheaf, each global section of \( \mathscr{F}(n) \) restricts a section \( \Gamma(D(f), \mathscr{F}) \). This gives a map \( (\Gamma(\mathscr{F}))^{\tilde{}} \rightarrow \mathscr{F} \) over \( D(f) \) for each \( f \in S _ 1 \), and they glue well. So there are canonical maps \[ \displaystyle \alpha : M \rightarrow \Gamma(\tilde{M}), \quad \beta : (\Gamma(\mathscr{F}))^{\tilde{}} \rightarrow \mathscr{F}. \]
Example 3. We have seen that \( \Gamma(\mathbb{P} _ A^n, \mathcal{O}(m)) = H^0(\mathbb{P} _ A^n, \mathcal{O}(m)) \) is isomorphic to the space of degree \( m \) homogeneous polynomials in \( S = A[x _ 0, \ldots, x _ n] \). So \( \alpha : S \rightarrow \Gamma(\mathcal{O} _ X) \) is an isomorphism.
We expect these two operations \( \Gamma \) and \( \tilde{} \) to be somewhat inverses to each other.
Proposition 5 (Hartshorne II.5.15). Let \( S \) be a graded ring that is finitely generated by \( S _ 1 \) over \( S _ 0 \), (this just means that \( S \) is a quotient of \( S _ 0[x _ 0, \ldots, x _ n] \)) and let \( X = \operatorname{Proj} S \). If \( \mathscr{F} \) is a quasi-coherent sheaf over \( X \), then \( \beta : \Gamma(\mathscr{F})^{\tilde{}} \rightarrow \mathscr{F} \) is an isomorphism.
Proof. Let \( f _ 1, \ldots, f _ n \in S _ 1 \) generate \( S \) over \( S _ 0 \). Working affine-locally, we only need to check that the map of modules are bijective. To show injectivity is to show that if a global section \( s \in \mathscr{F}(m) \) vanishes over \( D(f _ i) \), then \( s \otimes f _ i^c = 0 \) in \( \Gamma(X, \mathscr{F}(m+c)) \) for a sufficiently large \( c \). But on each chart \( D(f _ j) \), the section vanishing on \( D(f _ i) \) means that \( s \in \Gamma(D(f _ j), \mathscr{F}) \) will localize to \( 0 \) in \( \Gamma(D(f _ j), \mathscr{F}) _ {f _ i/f _ j} \). Then \( s \cdot (f _ i / f _ j)^c = 0 \) for all \( j \) and large enough \( c \), and this just means that \( s \otimes f _ i^c \) vanishes in \( \mathscr{F}(m+c) \). To show surjectivity is to show that every section of \( D(f _ i) \) actually comes from a global section of \( \mathscr{F}(m+c) \). This is just because \( s \in \Gamma(D(f _ i), \mathscr{F}(m)) \) restricts to an element of \( \Gamma(D(f _ i f _ j), \mathscr{F}(m)) = \Gamma(D(f _ j), \mathscr{F}(m)) _ {f _ i/f _ j} \), and so we can find \( s _ j \) such that \( s _ j \) and \( s \) glue well in \( \mathscr{F}(m+c) \) for large \( c \). We might worry about \( s _ j \) and \( s _ {j^\prime} \) not gluing well, but injectivity tells us that they will agree once we set \( c \) higher. ▨
What about \( \alpha : M \rightarrow \Gamma(\tilde{M}) \)? It can’t be that \( \alpha \) is also an isomorphism in most cases. Consider the class \( \mathcal{C} \subseteq \mathsf{GrMod} _ {S} \) consisting of modules \( M \) such that \( M _ n = 0 \) for all sufficiently large \( n \). This is a Serre class and so we can consider the "localization" \( \mathsf{GrMod} _ S / \mathcal{C} \). It is not difficult to see that \( \tilde{} : \mathsf{GrMod} _ S \rightarrow \mathsf{QCoh} _ {\operatorname{Proj} S} \) factors through \( \mathsf{GrMod} _ S / \mathcal{C} \), that is, \( \mathcal{C} \)-bijective graded modules induce isomorphic quasi-coherent sheaves.
Proposition 6 (Serre n63 Proposition 3). Let \( S = k[x _ 0, \ldots, x _ n] \) with \( k \) a field, and let \( M \) be a finitely generated graded module (or \( \mathcal{C} \)-isomorphic to one). Then \( \alpha : M \rightarrow \Gamma(\tilde{M}) \) is an \( \mathcal{C} \)-isomorphism. Moreover, \( H^i(\mathbb{P} _ k^n, \tilde{M}(\bullet)) \rightarrow 0 \) is an \( \mathcal{C} \)-isomorphism for \( i > 0 \).
Proof. By Hilbert’s syzygy theorem (here’s a nice proof written by Tom Lovering), there exists a finite graded resolution of \( M \) \[ \displaystyle 0 \rightarrow F^s \rightarrow F^{s-1} \rightarrow \cdots \rightarrow F^0 \rightarrow M \rightarrow 0 \] by finite rank free modules. (By free, I mean that each is isomorphic to \( \bigoplus _ i S(n _ i) \). All maps should preserve grading.) We’re going to prove this by induction on the minimal length of such a resolution. If \( M \) is free, we have already checked this in Example 2. For longer lengths, consider a short exact \( 0 \rightarrow N \rightarrow F \rightarrow M \rightarrow 0 \) where \( F \) is finite rank free and \( N \) has a shorter resolution than \( M \). We have a short exact sequence \( 0 \rightarrow \tilde{N}(\bullet) \rightarrow \tilde{F}(\bullet) \rightarrow \tilde{M}(\bullet) \rightarrow 0 \) of sheaves (note that twisting is exact), and then a long exact sequence \[ \displaystyle \cdots \rightarrow H^i(X, \tilde{N}(\bullet)) \rightarrow H^i(X, \tilde{F}(\bullet)) \rightarrow H^i(M, \tilde{M}(\bullet)) \rightarrow \cdots. \] But all \( H^i(X, \tilde{N}(\bullet)) \) and \( H^i(X, \tilde{F}(\bullet)) \) are \( \mathcal{C} \)-isomorphic to \( 0 \) by the induction hypothesis, except for \( i = 0 \) in which case they are \( \mathcal{C} \)-isomorphic to \( N \) and \( F \). This shows that \( 0 \rightarrow N \rightarrow F \rightarrow \Gamma(X, \tilde{M}) \rightarrow 0 \) is a \( \mathcal{C} \)-exact sequence and \( H^i(X, \tilde{M}(\bullet)) \) is \( \mathcal{C} \)-isomorphic to \( 0 \). Therefore \( \alpha \) is an \( \mathcal{C} \)-isomorphism. ▨
For a quasi-coherent sheaf \( \mathscr{F} \) on \( \mathbb{P} _ k^n \) (or any locally Noetherian scheme in general), let us say that it is coherent if on each affine, the sheaf corresponds to a finitely generated module. Denote by \( \mathsf{Coh} _ X \) the full subcategory of coherent sheaves. It is not hard to see that a finitely generated graded module \( S \) induces a coherent sheaf. Conversely, a coherent sheaf induces a \( \mathcal{C} \)-finitely generated graded module.
Proposition 7 (Serre n60 Theorem 2). If \( \mathscr{F} \) be a coherent sheaf on \( \mathbb{P} _ k^n \), then \( \Gamma(\mathscr{F}) \) is \( \mathcal{C} \)-isomorphic to a finitely generated \( k[x _ 0, \ldots, k _ n] \)-module.
Proof. If \( \mathscr{F} \) is a coherent sheaf on \( \mathbb{P} _ k^n \), it is generated by a finite number of sections over \( D(f _ i) \), and hence finitely generated by global sections of \( \mathscr{F}(n) \) for \( n \) sufficiently large. This gives a surjection \( \mathscr{L}^0 \rightarrow \mathscr{F} \rightarrow 0 \) of sheaves, where \( \mathscr{L}^0 \) is a finite direct sum of \( \mathscr{O} _ X(m) \). The kernel is also coherent, and hence we can continue it to an exact \( \mathscr{L}^1 \rightarrow \mathscr{L}^0 \rightarrow \mathscr{F} \) where \( \mathscr{L}^i \) are finite direct sums of \( \mathscr{O} _ X(m) \). If we let \( M \) to be the cokernel \( \Gamma(\mathscr{L}^1) \rightarrow \Gamma(\mathscr{L}^0) \rightarrow M \rightarrow 0 \), then \( M \) is clearly finitely generated, because \( \Gamma(\mathscr{L}^0) \) is. If we apply the \( \tilde{} \) construction, we get an exact \( \mathscr{L}^1 \rightarrow \mathscr{L}^0 \rightarrow \tilde{M} \rightarrow 0 \), and this shows that \( \tilde{M} \cong \mathscr{F} \). Applying \( \Gamma \) shows that \( \Gamma(\mathscr{F}) \cong \Gamma(\tilde{M}) \), and \( \Gamma(\tilde{M}) \) is \( \mathcal{C} \)-isomorphic to \( M \), which is finitely generated. ▨
Corollary 8. For \( S = k[x _ 0, \ldots, x _ n] \) and \( X = \operatorname{Proj} S \), the functors \( \tilde{} : \mathsf{FinGrMod} _ S / \mathcal{C} \rightarrow \mathsf{Coh} _ {X} \) and \( \Gamma : \mathsf{Coh} _ {X} \rightarrow \mathsf{FinGrMod} _ S / \mathcal{C} \) exhibit an equivalence of the two categories.
If you’re really interested in what the essential image of \( \Gamma : \mathsf{Coh} _ X \rightarrow \mathsf{GrMod} _ S \) is, Serre has the following proposition.
Proposition 9 (Serre n67 Proposition 9). Let \( S = k[x _ 0, \ldots, x _ n] \). The essential image of \( \Gamma : \mathsf{Coh} _ X \rightarrow \mathsf{GrMod} _ S \) consists of \( \mathcal{C} \)-finitely generated modules \( M \) satisfying:
- If \( m \in M \) such that \( x _ i m = 0 \) for all \( i \), then \( m = 0 \).
- If homogeneous elements \( m _ i \in M \) of the same degree satisfy \( x _ j m _ i = x _ i m _ j \) for all \( i, j \), then there exists an \( m \in M \) such that \( m _ i = x _ i m \).
9. The Hilbert polynomial and Euler characteristic#
We can now discuss one application of all the theory we have developed so far. Let \( k \) be a field, and consider a graded ring \( S \) such that \( S _ 0 = k \) and \( S \) is finitely generated by \( S _ 1 \) over \( S _ 0 = k \). As we have said, this corresponds to a closed subscheme of \( \mathbb{P} _ k^n \) for some \( n \). Let us write \( X = \operatorname{Proj} S \) as usual.
Proposition 10. If \( X = \operatorname{Proj} S \) as above, and \( \mathscr{F} \) is a coherent sheaf over \( X \), then each \( H^i(X, \mathscr{F}) \) is a finite-dimensional \( k \)-vector space.
Proof. For \( \mathscr{F} \) a coherent sheaf over \( X \), we know that \( \Gamma(\mathscr{F}) \) is \( \mathcal{C} \)-isomorphic to a finitely generated \( M \). Then \( \mathscr{F} \cong \tilde{M} \). If we have a short exact sequence \( 0 \rightarrow N \rightarrow F \rightarrow M \rightarrow 0 \) where \( F \) is a finitely rank free module, then we get an exact \( 0 \rightarrow \tilde{N} \rightarrow \tilde{F} \rightarrow \tilde{M} \) and thus a long exact \[ \displaystyle \cdots \rightarrow H^i(X, \tilde{N}) \rightarrow H^i(X, \tilde{F}) \rightarrow H^i(X, \tilde{M}) \rightarrow H^{i+1}(X, \tilde{N}) \rightarrow \cdots. \] We know that each \( H^i(X, \tilde{F}) \) is a finite-dimensional \( k \)-vector space, because we exactly know the dimensions from Example 2. So if each \( H^i(X, \tilde{N}) \) has finite dimension, then each \( H^i(X, \tilde{M}) \) also has finite dimension. Thus we can prove this by induction on the minimal length of a finite rank free resolution. (Again, we’re using Hilbert’s syzygy theorem.) ▨
So let us define the number \[ \displaystyle h^i(X, \mathscr{F}) = \dim _ k H^i(X, \mathscr{F}) \] for a coherent sheaf \( \mathscr{F} \). If \( 0 \rightarrow \mathscr{F} \rightarrow \mathscr{G} \rightarrow \mathscr{H} \rightarrow 0 \) is exact, then we get a long exact \[ \displaystyle \cdots \rightarrow H^i(X, \mathscr{F}) \rightarrow H^i(X, \mathscr{G}) \rightarrow H^i(X, \mathscr{H}) \rightarrow H^{i+1}(X, \mathscr{F}) \rightarrow \cdots. \] If we define the Euler characteristic of \( \mathscr{F} \) as \[ \displaystyle \chi(X, \mathscr{F}) = \sum _ {i=0}^{\infty} (-1)^i h^i(X, \mathscr{F}), \] (note that this is a finite sum because cohomology vanishes for \( i > n \)) it follows that \( \chi(X, \mathscr{F}) + \chi(X, \mathscr{H}) = \chi(X, \mathscr{G}) \).
Proposition 11. For any coherent \( \mathscr{F} \), the Euler characteristics \( \chi(X, \mathscr{F}(m)) \) is a polynomial in \( m \).
Proof. We again use induction on the length of the resolution. We first check that if \( \mathscr{F} = \mathscr{O} _ X \), then \[ \displaystyle \chi(X, \mathscr{O} _ X(m)) = \frac{(m+1) (m+2) \cdots (m+n)}{n!}. \] So \( \chi(X, \mathscr{L}(m)) \) is a polynomial in \( m \) if \( \mathscr{L} \) is a direct sum of a finite number of \( \mathscr{O} _ X(m _ i) \). Also, if \( 0 \rightarrow \mathscr{G} \rightarrow \mathscr{L} \rightarrow \mathscr{F} \rightarrow 0 \) is exact and \( \chi(X, \mathscr{G}(m)) \) is a polynomial, then \( \chi(X, \mathscr{F}(m)) = \chi(X, \mathscr{L}(m)) - \chi(X, \mathscr{G})) \) is also a polynomial. ▨
Definition 12. This polynomial \( p _ X(m) = \chi(X, \mathscr{O} _ X(m)) \) is called the Hilbert polynomial of \( m \).
Corollary 13. The dimension \( \dim _ k S _ m \) is equal to a polynomial in \( m \), namely \( p _ X(m) \), for \( m \) sufficiently large.
Proof. Note that we have proven that \( H^i(X, \tilde{M}(m)) = 0 \) and \( H^0(X, \tilde{M}(m)) \cong M _ m \) for \( m \) sufficiently large and \( i > 0 \). Let \( M = S \). ▨
Again, the Hilbert polynomial is not well-defined if we’re only given the scheme \( X \). Because twisting is defined when there is an embedding \( X \hookrightarrow \mathbb{P} _ k^n \) (or just a very ample line bundle), this data is needed to define the polynomial. But \( \mathscr{F}(0) = \mathscr{F} \) always, and so we can evaluate \( p _ X(0) \) without any trouble.
Definition 14. We define the Euler characteristic of \( X \) as \[ \displaystyle \chi(X) = \chi(X, \mathscr{O} _ X) = p _ X(0) = \sum _ {i = 0}^{\infty} (-1)^i h^i(X, \mathscr{O} _ X). \]
Corollary 15. The value of \( p _ X(0) \) doesn’t depend on the embedding \( X \hookrightarrow \mathbb{P} _ k^n \) (or, the very ample line bundle \( \mathscr{O} _ X(1) \)).
This can be used to define the arithmetic genus of a curve. If \( X \) is a curve and \( g \) is its genus, they will satisfy the formula \( \chi = 1 - g \). For instance, the Hilbert polynomial for \( \mathbb{P} _ k^1 \) is \( \chi _ {\mathbb{P} _ k^1}(m) = \frac{1}{2} (m+1) (m+2) \) and so \( \chi(\mathbb{P} _ k^1) = 1 \). For an elliptic curve \( E \hookrightarrow \mathbb{P} _ k^2 \), we have \( \chi _ E(m) = 3m \) and so \( \chi(E) = 0 \).
References#
[Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1997, Graduate Texts in Mathematics, No. 52.
[Ser55] Jean-Pierre Serre, Faisceaux Algébriques Cohérents, Ann. of Math. (2) 61 (1955), 197–278.