This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
We previously defined the line bundle \(\mathcal{O}(d)\) in [Algebraic Varieties] §Line Bundles and Vector Bundles, ⁋Example 16, and verified that its global sections \(H^0(\mathbb{P}^n, \mathcal{O}(d))\) are isomorphic to homogeneous polynomials of degree \(d\). However, the sheaf cohomology we introduced in the previous post ([Algebraic Varieties] §Sheaf Cohomology, ⁋Definition 1) is a richer invariant that includes not only \(H^0\) but also the higher cohomology groups \(H^1, H^2, \ldots\). Therefore, we now extract all information about \(\mathcal{O}(d)\) using not only \(H^0\) but also the higher cohomology groups.
Bott’s Formula
Since \(\mathcal{O}(d)\) is a line bundle, it is a quasi-coherent sheaf, and thus to compute sheaf cohomology it suffices to use Čech cohomology with the standard affine cover \(\mathcal{U}=\{U_0,\ldots, U_n\}\). The following is the result of that computation.
Proposition 1 (Bott) The cohomology of the line bundle \(\mathcal{O}(d)\) on \(\mathbb{P}^n\) is given by:
\[H^q(\mathbb{P}^n, \mathcal{O}(d)) = \begin{cases} \mathbb{K}[\x_0, \ldots, \x_n]_d & q = 0, d \geq 0 \\ \mathbb{K}[\x_0^{-1}, \ldots, \x_n^{-1}]_{-d-n-1} & q = n, d \leq -n-1 \\ 0 & \text{otherwise} \end{cases}\]Proof
As explained above, we use Čech cohomology. First recall that on each open set, the sections \(\mathcal{O}(d)(U_i)\) are
\[\x_i^d \cdot \mathbb{K}[\x_0/\x_i, \ldots, \widehat{\x_i/\x_i}, \ldots, \x_n/\x_i]\]([Algebraic Varieties] §Line Bundles and Vector Bundles, ⁋Example 12). Then a Čech cochain \(f \in \check{C}^p(\mathcal{U}, \mathcal{O}(d))\) assigns to each \((p+1)\)-tuple \((i_0, \ldots, i_p)\) a regular section over the open set \(U_{i_0}\cap\cdots\cap U_{i_p}\). For a section to be regular on the intersection \(U_{i_0}\cap\cdots\cap U_{i_p}\), only the non-vanishing coordinates, namely \(\x_{i_0}, \ldots, \x_{i_p}\), may appear in the denominator; the remaining coordinates may not. Thus the sections are generated by monomials of the form
\[f_{i_0 \cdots i_p} = \x_0^{a_0} \cdots \x_n^{a_n},\qquad \sum_{j=0}^n a_j=d,\quad a_j\geq 0\text{ for $j\not\in \{i_0, \ldots, i_p\}$}.\]The coboundary map \(\delta : \check{C}^p \to \check{C}^{p+1}\) is given by
\[(\delta f)_{i_0 \cdots i_{p+1}} = \sum_{k=0}^{p+1} (-1)^k f_{i_0 \cdots \hat{i_k} \cdots i_{p+1}}.\]This is the alternating sum of the sections corresponding to the \(p\)-tuples obtained by omitting one index from each \((p+1)\)-tuple.
Now let us use this data to compute the cohomology groups. Starting with the case of \(\mathbb{P}^1\), the Čech complex is
\[0 \longrightarrow \check{C}^0\overset{\delta}{\longrightarrow}\check{C}^1\longrightarrow 0\]where
\[\check{C}^0=\mathcal{O}(d)(U_0)\oplus \mathcal{O}(d)(U_1),\qquad \check{C}^1=\mathcal{O}(d)(U_0\cap U_1)\]and the respective section spaces are
\[\mathcal{O}(d)(U_0) = \x_0^d \cdot \mathbb{K}[\x_1/\x_0], \qquad \mathcal{O}(d)(U_1) = \x_1^d \cdot \mathbb{K}[\x_0/\x_1], \qquad \mathcal{O}(d)(U_0 \cap U_1) = \mathbb{K}[\x_0^{\pm 1}, \x_1^{\pm 1}]_d.\]First, to compute the cohomology of \(\check{C}^0\), we analyze \(\ker\delta\). Since \(H^0(\mathbb{P}^n, \mathcal{O}(d))=\Gamma(\mathbb{P}^n, \mathcal{O}(d))\), this is essentially a recheck of the computation in [Algebraic Varieties] §Line Bundles and Vector Bundles, ⁋Example 16, but instead of treating it as a separate example, we perform the Čech cohomology calculation within this proof.
By definition, a cochain \((f_0, f_1) \in \check{C}^0\) lies in \(\ker \delta\) if and only if \(f_0 = f_1\) in \(\mathcal{O}(d)(U_0 \cap U_1)\). Looking at the \(U_0\) part first, we know that any monomial in \(\mathcal{O}(d)(U_0)\) must be of the form \(\x_0^{d-a}\x_1^a\) for some \(a\geq 0\). Similarly, any monomial in \(\mathcal{O}(d)(U_1)\) must be of the form \(\x_0^b\x_1^{d-b}\) for some \(b\geq 0\). Now for a specific cocycle \((f_0,f_1)\) to lie in \(\ker\delta\), we need \(f_0=f_1\), and therefore only monomials satisfying \(a+b=d\) can belong to \(\ker\delta\). That is, the monomials
\[\x_0^d, \quad\x_0^{d-1}\x_1,\quad\ldots, \quad\x_0\x_1^{d-1},\quad \x_1^d\]form a basis of \(H^0\), which gives the desired result. If \(d<0\), then \(a,b\geq 0\) cannot satisfy this equation, so \(H^0\) is \(0\).
Now let us compute \(H^1\). We must compute \(\coker\delta\). From the above computation, we know that the image of \(\delta\) consists of elements of the form
\[f_1-f_0=\sum_{i\geq 0}a_i \x_0^{d-i}\x_1^i-\sum_{j\geq 0}b_j\x_0^j\x_1^{d-j}\tag{$\ast$}\]for suitable constants \(a_i,b_j\). On the other hand, elements of \(\check{C}^1\) are generated by degree-\(d\) monomials
\[\x_0^{2d}\x_1^{-d},\quad,\x_0^{2d-1}\x_1^{-d+1}, \quad, \ldots,\quad \x_0^{-d+1}\x_1^{2d-1},\quad\x_0^{-d}\x_1^{2d}\tag{$\ast\ast$}\]If \(d\geq 0\), then each of these can be obtained explicitly from (\(\ast\)). For instance, \(\x_0^{2d}\x_1^{-d}\) comes from the \(j=2d\) term in the \(f_1\) component, and \(\x_0^{-d}\x_1^{2d}\) comes from the \(i=2d\) term in the \(f_0\) component. Hence in this case \(\coker\delta=0\). However, if \(d<0\), there arise monomials that cannot be represented in the image of \(\delta\), because analyzing the terms in (\(\ast\)) shows that at least one exponent is greater than or equal to \(0\). On the other hand, in (\(\ast\ast\)) there appear monomials where both exponents are negative:
\[\x_0^{-1}\x_1^{d+1}, \quad \x_0^{-2}\x_1^{d+2},\quad,\ldots, \x_0^{d+1}\x_1^{-1}\]and these generate \(\coker \delta\). We will explain the notation in the statement separately after the proof.
Now let us use induction for the general case. To this end, using the fact that the hyperplane \(H=\{\x_n=0\}\) in \(\mathbb{P}^n\) is isomorphic to \(\mathbb{P}^{n-1}\), consider the short exact sequence
\[0 \longrightarrow \mathcal{O}(d-1)\overset{\times \x_n}{\longrightarrow} \mathcal{O}(d)\longrightarrow \mathcal{O}(d)\vert_H\longrightarrow 0.\]From this we obtain the long exact sequence
\[\cdots \to H^{i-1}(\mathbb{P}^{n-1}, \mathcal{O}(d)) \to H^i(\mathbb{P}^n, \mathcal{O}(d-1)) \to H^i(\mathbb{P}^n, \mathcal{O}(d)) \to H^i(\mathbb{P}^{n-1}, \mathcal{O}(d)) \to \cdots\]and by the inductive process we already know the claim for \(\mathbb{P}^{n-1}\). Then in particular for \(0<i<n\) we have
\[H^{i-1}(\mathbb{P}^{n-1}, \mathcal{O}(d))=H^i(\mathbb{P}^{n-1}, \mathcal{O}(d))=0\]and from this we obtain \(H^i(\mathbb{P}^n, \mathcal{O}(d-1)) \cong H^i(\mathbb{P}^n, \mathcal{O}(d))\). That is, \(H^i(\mathbb{P}^n, \mathcal{O}(d))\) is isomorphic for all \(d\), and in particular we can immediately show that the Čech cohomology for the convenient sheaf \(\mathcal{O}\)
\[H^i(\mathbb{P}^n, \mathcal{O}(d))=H^i(\mathbb{P}^n, \mathcal{O})=0\]and therefore we know that
\[H^i(\mathbb{P}^n, \mathcal{O}(d))=0\]holds for all \(0<i<n\) and all \(d\).
Now consider the top cohomology part
\[\cdots\rightarrow H^{n-1}(\mathbb{P}^{n-1}, \mathcal{O}(d))\rightarrow H^n(\mathbb{P}^n, \mathcal{O}(d-1))\rightarrow H^n(\mathbb{P}^n, \mathcal{O}(d))\rightarrow H^n(\mathbb{P}^{n-1}, \mathcal{O}(d))=0.\]If \(d\geq -n\), then again by the inductive hypothesis \(H^{n-1}(\mathbb{P}^{n-1}, \mathcal{O}(d))=0\), so by the same argument we can show that \(H^n(\mathbb{P}^n, \mathcal{O}(d))\) holds for all \(d>-n-1\). In the case \(d\leq -n-1\), \(H^n\) must be computed directly from the Čech complex; computing \(\check{C}^n(\mathbb{P}^n, \mathcal{O}(d))\) directly, we know that an \(n\)-cochain is an element of
\[\mathbb{K}[\x_0^{\pm 1}, \ldots, \x_n^{\pm 1}]_d\]and the monomials not appearing in the image of an \((n-1)\)-cochain are, similarly to the computation for \(\mathbb{P}^1\) above, degree-\(d\) monomials with all exponents less than \(-1\), from which we obtain the desired result. The case of \(H^0\) was computed directly above, but as already mentioned, this is merely a recheck of [Algebraic Varieties] §Line Bundles and Vector Bundles, ⁋Example 16, so we will not repeat it here.
In the above proof we saw that for each variable \(\x_0,\cdots, \x_n\) and \(d\leq -n-1\), the group \(H^n(\mathbb{P}^n, \mathcal{O}(d))\) is generated by monomials of the form
\[\x_0^{a_0} \cdots \x_n^{a_n},\qquad a_i \leq -1, \quad \sum a_i=d.\](Note that \(d\) is negative.) If we think of each \(\x_i^{-1}\) as a new variable \(\y_i=\x_i^{-1}\), this is the space generated by expressions of the form
\[\y_0^{\lvert a_0\rvert},\cdots \y_n^{\lvert a_n\rvert}\qquad \lvert a_i\rvert\geq 1,\quad \sum \lvert a_i\rvert=\lvert d\rvert.\]Here, since each \(a_i\) and \(d\) are negative, we have \(\lvert a_i\rvert=-a_i\) and \(\lvert d\rvert=-d\). This space is almost like the space of homogeneous polynomials of degree \(\lvert d\rvert\), except that the \(\lvert a_i\rvert\) cannot be \(0\). Therefore, substituting \(b_i=\lvert a_i\rvert-1\), we can think of this space as the space of expressions
\[\y_0^{b_i}\cdots \y_n^{b_n},\qquad b_i\geq 0,\quad \sum b_i=\lvert d\rvert-(n+1).\]That is, this space can be thought of as the space of “negative degree” monomials of degree \(-d-n-1\), and for this reason this space is denoted by
\[\mathbb{K}[\x_0^{-1}, \ldots, \x_n^{-1}]_{-d-n-1}.\]For later use we define the Euler characteristic.
Definition 2 For a variety \(X\) and a coherent sheaf \(\mathcal{F}\) defined on it, the Euler characteristic of \(\mathcal{F}\) is defined by the formula
\[\rchi(X, \mathcal{F}) = \sum_{i=0}^{n} (-1)^i \dim H^i(X, \mathcal{F}).\]In the special case where \(X=\mathbb{P}^n\) and \(\mathcal{F}=\mathcal{O}(d)\), in any case the intermediate cohomology groups all vanish and we only need to consider the cohomology at the two ends, so we can easily prove the following corollary.
Corollary 3 The Euler characteristic of \(\mathcal{O}(d)\) on \(\mathbb{P}^n\) is given by the formula
\[\rchi(\mathbb{P}^n, \mathcal{O}(d)) = \binom{n+d}{n}.\]Proof
By Proposition 1, the cohomology falls into three cases.
First, if \(d \geq 0\), only \(H^0\) is non-zero, so
\[\rchi(\mathcal{O}(d)) = \dim H^0(\mathbb{P}^n, \mathcal{O}(d)) = \dim \mathbb{K}[\x_0, \ldots, \x_n]_d = \binom{n+d}{n}\]holds.
Second, if \(-n \leq d \leq -1\), all cohomology vanishes, so \(\rchi(\mathcal{O}(d)) = 0\), and in this case we usually define \(\binom{n+d}{n}=0\), which agrees with the convention.
Finally, consider the case \(d \leq -n-1\). In this case, only \(H^n\) is non-zero, so
\[\rchi(\mathcal{O}(d)) = (-1)^n \dim \mathbb{K}[\x_0^{-1}, \ldots, \x_n^{-1}]_{-d-n-1}.\]By the explanation immediately after Proposition 1, the dimension of this space is
\[\binom{-d-1}{n}=(-1)^n\binom{n+d}{n}.\]Here \(\binom{n+d}{n}\) follows the general convention for binomial coefficient notation, as in the previous case.
The Euler characteristic has the important property of additivity for short exact sequences. That is, for a short exact sequence
\[0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0\]we have \(\rchi(\mathcal{G}) = \rchi(\mathcal{F}) + \rchi(\mathcal{H})\). Therefore, the Euler characteristic becomes an invariant that is much easier to compute and manipulate, at the cost of losing information about individual cohomology groups.
Serre Vanishing
According to Proposition 1, on \(\mathbb{P}^n\) the higher cohomology of \(\mathcal{O}(d)\) vanishes for sufficiently large \(d\). Since any line bundle on \(\mathbb{P}^n\) is of the form \(\mathcal{O}(d)\) for some \(d\), this means that for any line bundle \(\mathcal{L}\) on \(\mathbb{P}^n\) and sufficiently large \(d\gg 0\), the twisted line bundle
\[\mathcal{L}\otimes \mathcal{O}(d)\]necessarily has higher cohomology equal to \(0\).
More generally, we can extend this to arbitrary projective varieties and arbitrary coherent sheaves defined on them. For this we first need something to play the role of \(\mathcal{O}(1)\); in our definition, a projective variety \(X\) is always given by an embedding \(X\hookrightarrow\mathbb{P}^N\), so we can pull back \(\mathcal{O}(1)\) from \(\mathbb{P}^N\).
Proposition 4 (Serre Vanishing) Let \(X\) be a projective variety, \(\mathcal{L}\) an ample line bundle, and \(\mathcal{F}\) a coherent sheaf. Then for sufficiently large \(m\),
\[H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m}) = 0 \quad (i > 0)\]holds.
Proof
Since \(\mathcal{L}\) is ample, for sufficiently large \(m_0\) the sheaf \(\mathcal{L}^{\otimes m_0}\) is very ample. That is, there exists a suitable embedding \(i \colon X \hookrightarrow \mathbb{P}^N\) such that \(\mathcal{L}^{\otimes m_0} = i^*\mathcal{O}(1)\). Restricting the standard affine cover \(\{U_i\}\) of \(\mathbb{P}^N\) to \(X\) gives an affine open cover \(\{X \cap U_i\}\). Since a finite intersection \(U_{i_0} \cap \cdots \cap U_{i_p}\) is affine, \((X \cap U_{i_0}) \cap \cdots \cap (X \cap U_{i_p}) = X \cap (U_{i_0} \cap \cdots \cap U_{i_p})\) is also affine. Therefore, since the two Čech complexes are literally the same,
\[\check{H}^i(\{X \cap U_j\}, \mathcal{F}) = \check{H}^i(\{U_j\}, i_*\mathcal{F})\]holds. Since \(X\) and \(\mathbb{P}^N\) are separated schemes ([Algebraic Varieties] §Sheaf Cohomology, ⁋Theorem 11), for quasi-coherent sheaves Čech cohomology equals sheaf cohomology:
\(H^i(X, \mathcal{F}) = \check{H}^i(\{X \cap U_j\}, \mathcal{F}) = \check{H}^i(\{U_j\}, i_*\mathcal{G}) = H^i(\mathbb{P}^N, i_*\mathcal{G}).\) Therefore it suffices to show the following: for a coherent sheaf \(\mathcal{G}\) on \(\mathbb{P}^N\), for sufficiently large \(n\) we have \(H^i(\mathbb{P}^N, \mathcal{G}(n)) = 0\) (\(i > 0\)). Here \(\mathcal{G}(n) = \mathcal{G} \otimes \mathcal{O}_{\mathbb{P}^N}(n)\).
Key Lemma. We show that \(\mathcal{G}(n)\) is globally generated for sufficiently large \(n\). (See ⁋Definition 6 below.)
Let \(S = \mathbb{K}[\x_0, \ldots, \x_N]\), and let \(M = \bigoplus_{n \in \mathbb{Z}} \Gamma(\mathbb{P}^N, \mathcal{G}(n))\) be the associated graded \(S\)-module. On each standard affine open set \(D_+(\x_j)\), the sections \(\Gamma(D_+(\x_j), \mathcal{G})\) are the degree-\(0\) localization \(M_{(\x_j)}\), which is a finitely generated module over \(S_{(\x_j)}\). Choose generators \(\bar{m}_1, \ldots, \bar{m}_{r_j} \in M_{(\x_j)}\). Each \(\bar{m}_k\) can be written in the form \(m_k / \x_j^{d_k}\), where \(m_k \in M\) is a homogeneous element. Letting \(d_0 = \max_j \max_k d_k\), we multiply each generator by \(\x_j^{d_0 - d_k}\) to obtain homogeneous elements \(m_k \cdot \x_j^{d_0 - d_k} \in M_{d_0}\). These are elements of \(\Gamma(\mathbb{P}^N, \mathcal{G}(d_0))\), and one can check that they generate the stalk of \(\mathcal{G}\) on \(D_+(\x_j)\). Taking the maximum over \(j\), we obtain that \(\mathcal{G}(d_0)\) is globally generated.
Vanishing. We now show \(H^i(\mathbb{P}^N, \mathcal{G}(n)) = 0\) (\(i > 0\), \(n \gg 0\)).
If \(N = 0\), then \(\mathbb{P}^0\) is a point, so this is trivial. Assume \(N \geq 1\). By the above lemma, \(\mathcal{G}(n_0)\) is globally generated (\(n_0 \gg 0\)), so there exists a suitable surjection
\[\mathcal{O}_{\mathbb{P}^N}^{\oplus r_0} \twoheadrightarrow \mathcal{G}(n_0).\]The kernel \(\mathcal{K}_0\) is coherent. From the long exact sequence of the short exact sequence
\[0 \to \mathcal{K}_0 \to \mathcal{O}^{\oplus r_0} \to \mathcal{G}(n_0) \to 0,\]since \(H^j(\mathbb{P}^N, \mathcal{O}^{\oplus r_0}) = 0\) (\(j > 0\)) by Proposition 1, we obtain
\[H^j(\mathcal{G}(n_0)) \cong H^{j+1}(\mathcal{K}_0) \quad (j \geq 1).\]We now repeat the same process for \(\mathcal{K}_0\). That is, choose \(n_1 \gg 0\) such that \(\mathcal{K}_0(n_1)\) is globally generated, and for the kernel \(\mathcal{K}_1\) of the surjection
\[\mathcal{O}^{\oplus r_1} \twoheadrightarrow \mathcal{K}_0(n_1)\]from the long exact sequence of
\[0 \to \mathcal{K}_1 \to \mathcal{O}^{\oplus r_1} \to \mathcal{K}_0(n_1) \to 0\]we obtain
\[H^{j+1}(\mathcal{K}_0(n_1)) \cong H^{j+2}(\mathcal{K}_1) \quad (j \geq 1).\]Repeating this process \(N\) times, we get
\[H^j(\mathcal{G}(n_0)) \cong H^{j+N}(\mathcal{K}_{N-1}).\]Since the cohomological dimension of \(\mathbb{P}^N\) is \(N\), we have \(H^{j+N} = 0\) (\(j \geq 1\), \(j + N \geq N+1 > N\)), and therefore \(H^j(\mathcal{G}(n_0)) = 0\).
Finally, since \(\mathcal{G}(n_0)\) is globally generated, \(\mathcal{G}(n) = \mathcal{G}(n_0) \otimes \mathcal{O}(n - n_0)\) is also globally generated for \(n \geq n_0\), and thus the same resolution argument can be constructed for \(\mathcal{G}(n)\), so the vanishing holds for all \(n \geq n_0\).
Regularity
Proposition 4 gave a qualitative result that higher cohomology vanishes after sufficiently large twisting. Regularity quantifies this, measuring exactly how much twisting is needed.
Intuitively, higher cohomology arises from failures in lower-degree cohomology, so this twisting is “less” necessary in high degrees. Keeping this in mind, the following definition is natural.
Definition 5 Let a projective variety \(X\) and an ample line bundle \(\mathcal{L}\) on it be fixed. Then a coherent sheaf \(\mathcal{F}\) on \(X\) is said to be \(m\)-regular if for all \(i>0\),
\[H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m - i}) = 0\]holds.
In general, computing all cohomology groups of a coherent sheaf is almost impossible, but the basic idea is that sufficiently twisting makes the higher cohomology vanish. Regularity goes further: it is the concept that measures exactly how much twisting is needed.
One of the key reasons such twisting is useful when dealing with coherent sheaves is that sufficiently twisting makes a coherent sheaf globally generated. To gain intuition for this concept, let us first think about the case of line bundles. A line bundle \(\mathcal{L}\) being basepoint-free, as defined in [Algebraic Varieties] §Linear Systems, ⁋Definition 5, means that for every point \(p \in X\) there exists a global section \(s \in H^0(X, \mathcal{L})\) with \(s(p) \neq 0\). That is, the base locus is empty, and the linear system \(\lvert \mathcal{L} \rvert\) provides a non-zero value at each point. This is equivalent to the evaluation map
\[H^0(X, \mathcal{L}) \otimes \mathcal{O}_X \to \mathcal{L}\]being surjective. Globally generated generalizes this condition to arbitrary coherent sheaves: a coherent sheaf \(\mathcal{F}\) is globally generated if, similarly, the evaluation map of the above form is surjective, so that the stalk at each point can be generated by global sections. In particular, for line bundles, being globally generated is equivalent to being basepoint-free. This property played a key role in the proof of Proposition 4.
Definition 6 A coherent sheaf \(\mathcal{F}\) is said to be globally generated if the evaluation map
\[H^0(X, \mathcal{F}) \otimes \mathcal{O}_X \to \mathcal{F}\]is surjective. That is, the stalks can all be generated by global sections.
To define regularity in general, we first need the concept of twist. On \(\mathbb{P}^n\) we use \(\mathcal{O}(1)\) as the base, so we write \(\mathcal{F}(d) := \mathcal{F} \otimes \mathcal{O}(d)\). On an arbitrary projective variety \(X\), we choose an ample line bundle \(\mathcal{L}\) and define \(\mathcal{F}(d) := \mathcal{F} \otimes \mathcal{L}^{\otimes d}\). Twist satisfies the following properties. By the associativity of the tensor product, \(\mathcal{F}(d)(e) = \mathcal{F}(d+e)\) holds. Also, since the tensor product functor \(- \otimes \mathcal{L}^{\otimes d}\) is a line bundle, it is exact, and therefore for a short exact sequence
\[0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0\]the sequence
\[0 \to \mathcal{F}(d) \to \mathcal{G}(d) \to \mathcal{H}(d) \to 0\]is also a short exact sequence.
Proposition 7 (Castelnuovo-Mumford Regularity) Let \(X\) be a projective variety, \(\mathcal{L}\) an ample line bundle, and \(\mathcal{F}\) a coherent sheaf. If \(\mathcal{F}\) is \(m\)-regular with respect to \(\mathcal{L}\), then the following hold.
- \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated.
- \(\mathcal{F} \otimes \mathcal{L}^{\otimes p}\) is \((m+p)\)-regular with respect to \(\mathcal{L}\) for all \(p \geq 0\).
Proof
We prove by induction on the dimension of \(X\). If \(\dim X = 0\), then \(X\) is a point and a coherent sheaf \(\mathcal{F}\) is a finite-dimensional vector space, so all cohomology except \(H^0\) automatically vanishes. Now assume \(\dim X \geq 1\).
The key is to use the restriction exact sequence for the effective divisor \(D\) defined by a global section \(s \in H^0(X, \mathcal{L})\) of \(\mathcal{L}\). Choosing a general \(s\), by Bertini’s theorem \(D\) is smooth, and we obtain the following short exact sequence.
\[0 \to \mathcal{F} \otimes \mathcal{L}^{\otimes k-1} \xrightarrow{\cdot s} \mathcal{F} \otimes \mathcal{L}^{\otimes k} \to \mathcal{F} \otimes \mathcal{L}^{\otimes k}\vert_D \to 0\]The cohomology long exact sequence of this sequence gives the following.
\[\cdots \to H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes k-1}) \to H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes k}) \to H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes k}\vert_D) \to H^{i+1}(\mathcal{F} \otimes \mathcal{L}^{\otimes k-1}) \to \cdots\]In the special case of \(\mathbb{P}^n\), we have \(\mathcal{L} = \mathcal{O}(1)\), \(s\) is a general linear form, and \(D\) becomes a hyperplane \(H\) isomorphic to \(\mathbb{P}^{n-1}\).
Step 1: \(m\)-regularity of \(\mathcal{F}\vert_D\). Since \(\mathcal{F}\) is \(m\)-regular with respect to \(\mathcal{L}\), we have \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i}) = 0\) for \(i > 0\). Let us show that \(\mathcal{F}\vert_D\) is \(m\)-regular with respect to \(\mathcal{L}\vert_D\). Substituting \(k = m - i\) in the restriction sequence (\(0 < i \leq n-1\)),
\[0 \to \mathcal{F} \otimes \mathcal{L}^{\otimes m-i-1} \to \mathcal{F} \otimes \mathcal{L}^{\otimes m-i} \to \mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m-i} \to 0\]and from its long exact sequence,
\[H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i}) \to H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m-i}) \to H^{i+1}(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i-1}).\]By \(m\)-regularity, \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i}) = 0\), and \(H^{i+1}(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i-1}) = 0\) (\(i+1 > 0\)), so we obtain
\[H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m-i}) = 0\]for \(0 < i \leq n-1\). This means that \(\mathcal{F}\vert_D\) is \(m\)-regular with respect to \(\mathcal{L}\vert_D\).
Step 2: \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated. Apply the inductive hypothesis to \(D\). \(D\) is a projective variety with \(\dim D < \dim X\), and \(\mathcal{L}\vert_D\) is an ample line bundle. Since \(\mathcal{F}\vert_D\) is \(m\)-regular, by the inductive hypothesis \(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m}\) is globally generated on \(D\).
Now let us show that \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated. It suffices to check that the fiber \((\mathcal{F} \otimes \mathcal{L}^{\otimes m})_p\) at an arbitrary point \(p \in X\) is generated by the images of global sections. Choose a general divisor \(D\) passing through \(p\), and substitute \(k = m\) in the restriction sequence:
\[0 \to \mathcal{F} \otimes \mathcal{L}^{\otimes m-1} \to \mathcal{F} \otimes \mathcal{L}^{\otimes m} \to \mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m} \to 0.\]From \(m\)-regularity with \(i = 1\), we have \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m-1}) = 0\), so
\[H^0(\mathcal{F} \otimes \mathcal{L}^{\otimes m}) \to H^0(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m})\]is surjective. By the inductive hypothesis, \(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m}\) is globally generated on \(D\), so its fiber at \(p\) is generated by the image of \(H^0(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m})\). Since the restriction map is surjective, the global sections of \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) also generate the fiber at \(p\). Therefore \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated.
Step 3: \(\mathcal{F} \otimes \mathcal{L}^{\otimes p}\) is \((m+p)\)-regular. Since \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated, there exists a surjection
\[\mathcal{O}_X^{\oplus r_0} \twoheadrightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m}\]for suitable \(r_0\). Tensoring this with \(\mathcal{L}^{\otimes p}\) gives
\[\mathcal{L}^{\oplus r_0} \twoheadrightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m+p}.\]Therefore, if \(H^i(X, \mathcal{L}^{\otimes p}) = 0\) for arbitrary \(i > 0\) and \(p \geq 0\), then \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p}) = 0\) holds. For \(p = 0\), the vanishing \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m}) = 0\) (\(i > 0\)) is exactly the definition of \(m\)-regularity of \(\mathcal{F}\). For \(p \geq 1\), since \(\mathcal{L}\) is ample, Proposition 4 implies that \(H^i(\mathcal{L}^{\otimes p}) = 0\) for sufficiently large \(p\), but for small \(p\) this factor may not vanish.
To resolve this issue, we use induction on \(p\). For \(p = 0\), the fact that \(\mathcal{F}(m)\) is \(m\)-regular is the definition. Assume \(p \geq 1\), and let us show that \(\mathcal{F}(m+p)\) is \((m+p)\)-regular, i.e., \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i}) = 0\) (\(i > 0\)). For \(i = 1\), substituting \(k = m + p - 1\) in the restriction sequence gives
\[H^0(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1}) \to H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-2}) \to H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-1}).\]By the inductive hypothesis (for \(p-1\)), we have \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-2}) = 0\). Also, since \(\mathcal{F}\vert_D\) is \(m\)-regular (Step 2), by the inductive hypothesis on dimension \(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes p}\) is \((m+p)\)-regular, and therefore \(H^1(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1}) = 0\). In the exact sequence, \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-1})\) embeds into \(H^1(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1})\), so we obtain \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-1}) = 0\). For \(i \geq 2\), from the same restriction sequence we have
\[H^{i-1}(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-i}) \to H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i-1}) \to H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i}) \to H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-i}).\]By the inductive hypothesis, \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i-1}) = 0\) (hypothesis for \(p' = p-1\), \(j = i\)), and by the inductive hypothesis for \(\mathcal{F}\vert_D\) (induction on dimension), \(H^{i-1}(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-i}) = 0\) and \(H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-i}) = 0\) hold for \(i-1 \geq 1\), \(i \leq n-1\). Therefore we obtain \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i}) = 0\).
Example 8 Let us compute the regularity of the line bundle \(\mathcal{O}(d)\) on \(\mathbb{P}^n\). Here \(\mathcal{L} = \mathcal{O}(1)\), so twisting is \(\mathcal{O}(d) \otimes \mathcal{O}(m) = \mathcal{O}(d+m)\). The \(m\)-regularity condition is \(H^i(\mathbb{P}^n, \mathcal{O}(d+m-i)) = 0\) (\(i > 0\)). If \(d \geq 0\) and we choose \(m = 0\), we need to check \(H^i(\mathcal{O}(d-i))\); for \(i = 1\), \(H^1(\mathcal{O}(d-1))\) is \(0\) if \(d \geq 1\), and if \(d = 0\) then \(H^1(\mathcal{O}(-1)) = 0\) (by Bott’s formula, since \(-1 \geq -n\), all cohomology is \(0\)). In general, if \(d \geq 0\) and \(i > 0\), then \(d - i \geq -n\) implies \(H^i(\mathcal{O}(d-i)) = 0\), and if \(d - i < -n\), i.e., \(i > d + n\), then \(i > n\) so anyway \(H^i = 0\). Therefore \(\mathcal{O}(d)\) is \(0\)-regular with respect to \(\mathcal{L} = \mathcal{O}(1)\). On the other hand, if \(d < 0\), then \(\mathcal{O}(d)\) is \((-d)\)-regular. By Proposition 7, \(\mathcal{O}(d) \otimes \mathcal{L}^{\otimes 0} = \mathcal{O}(d)\) is globally generated when \(d \geq 0\), which agrees with what we checked in ([Algebraic Varieties] §Line Bundles and Vector Bundles, ⁋Example 16).
Properties of Very Ample and Ample
The above Proposition 4 and Proposition 7 are representative results on the properties of ample line bundles. We conclude this post by examining additional properties of ample and very ample line bundles.
Proposition 9 If \(\mathcal{L}\) is very ample and \(\mathcal{M}\) is globally generated, then \(\mathcal{L} \otimes \mathcal{M}\) is very ample.
Proof
Since \(\mathcal{L}\) is very ample, there exists a projective embedding \(i: X \hookrightarrow \mathbb{P}^N\) such that \(\mathcal{L} = i^*\mathcal{O}_{\mathbb{P}^N}(1)\). On the other hand, since \(\mathcal{M}\) is globally generated, there exist global sections \(s_0, \ldots, s_n \in H^0(X, \mathcal{M})\) that generate the stalk at every point, and from these we can define a morphism \(\phi: X \to \mathbb{P}^n\).
Now consider the closed embedding \((i, \phi): X \to \mathbb{P}^N \times \mathbb{P}^n\). Composing this with the Segre embedding ([Algebraic Varieties] §Projective Varieties, ⁋Example 16)
\[\sigma: \mathbb{P}^N \times \mathbb{P}^n \hookrightarrow \mathbb{P}^{Nn+N+n}\]we have \(\sigma^*\mathcal{O}(1) = \pi_1^*\mathcal{O}(1) \otimes \pi_2^*\mathcal{O}(1)\), and therefore
\[(\sigma \circ (i, \phi))^*\mathcal{O}(1) = i^*\mathcal{O}(1) \otimes \phi^*\mathcal{O}(1) = \mathcal{L} \otimes \mathcal{M}.\]That is, \(\mathcal{L} \otimes \mathcal{M}\) is very ample.
That is, although the explanation was somewhat involved, the key point is that the morphism \(\phi:X\rightarrow \mathbb{P}^n\) defined by a globally generated line bundle \(\mathcal{M}\) may not be a closed embedding, but by tensoring with \(\mathcal{L}\) and embedding into projective space in the form \((i,\phi)\), the first component \(i\) makes this map a closed embedding. From this, the following useful result can also be proved.
Proposition 10 For an ample line bundle \(\mathcal{L}\) and an arbitrary line bundle \(\mathcal{M}\) defined on a projective variety \(X\), the sheaf \(\mathcal{M} \otimes \mathcal{L}^{\otimes n}\) is very ample for sufficiently large \(n\).
Proof
First, since \(\mathcal{L}\) is ample, for some \(m>0\) the sheaf \(\mathcal{L}^{\otimes m}\) is very ample. On the other hand, by Proposition 4 we can make the higher cohomology of \(\mathcal{M}\otimes \mathcal{L}^{\otimes k}\) vanish for sufficiently large \(k\), so for such \(k\) the sheaf \(\mathcal{M}\otimes \mathcal{L}^{\otimes k}\) is globally generated. Now by Proposition 9,
\[(\mathcal{M} \otimes \mathcal{L}^{\otimes k}) \otimes \mathcal{L}^{\otimes m} = \mathcal{M} \otimes \mathcal{L}^{\otimes (k+m)}\]is very ample, and setting \(n = k + m\) completes the proof.
References
[Har] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977.
[Bot] R. Bott, Homogeneous vector bundles, Annals of Mathematics, 1957.
[Laz] R. Lazarsfeld, Positivity in Algebraic Geometry I, Ergebnisse der Mathematik, Springer, 2004.
[Mum] D. Mumford, Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, Princeton, 1966.
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