Divisors and Vanishing Theorems

Some Homological Algebra

We first recall some basic facts in homological algebra:

Let $X$ be a variety, $\mathcal{F}^{\bullet}$ be the complex of coherent $\mathcal{O}_{X}$-module. The cohomology is defined by $\mathcal{H}^{i}(\mathcal{F}^{\bullet})=\frac{\mathop{\mathrm{ker}}d_{i}}{\mathop{\mathrm{im}}d_{i-1}}$, where $d$ is the differential.

A spectral sequence is a colletion of data: $(E_{r}^{p,q}, E^{n})$ and differential $d_{r}^{p,q}:E_{r}^{p,q}\to E_{r}^{p+r,q-r+1}$ such that $d_{r}^{p+r,q-r+1}\circ d_{r}^{p,q}=0$. So $d_{r}$ actually defines complexes. Moreover, we require $E_{r+1}^{p,q}=H^{0}(E_{r}^{p+\bullet r,q-\bullet r+1})$. For fixed $p,q$, if there is a number $r_{0}$ such that $d_{r}^{p,q}=d_{r}^{p-r,q+r-1}$ for all $r\geq r_{0}$, then the $E_{r}^{p,q}$ finally stabilizes, we say $E_{\infty}^{p,q}=E_{r}^{p,q}$ the limiting term. There exists a decreasing filtration $\cdots F^{p+1}E^{n}\subseteq F^{p}E^{n}\subseteq \cdots\subseteq E^{n}$ (in the context we may assume $\cap F^{p}E^{n}=0$ and $\cup F^{p}E^{n}=E^{n}$) such that $E_{\infty}^{p,q}=F^{p}E^{p+q} /F^{p+1}E^{p+q}$. Write

$$E_{r}^{p,q}\Rightarrow E^{p+q}.$$

A filtered complex $(C^{\bullet}, d)$ is a collection of subobjects $F^{p}C^{q}$ such that $F^{p+q}C^{q}\subseteq F^{p}C^{q+1}$ and $dF^{p}C^{q}\subseteq F^{p}C^{q+1}$ (n the context we may assume $\cap F^{p}C^{q}=0$ and $\cup F^{p}C^{q}=C^{q}$). Let

$$Z_{r}^{p,q}=\frac{d^{-1}(F^{p+r}C^{p+q+1})\cap F^{p}C^{p+q}}{d^{-1}(F^{p+r}C^{p+q+1})\cap F^{p+1}C^{p+q}};$$

$$B_{r}^{p,q}=\frac{d(F^{p-r+1}C^{p+q-1})\cap F^{p}C^{p+q}}{d(F^{p-r+q}C^{p+q})\cap F^{p+q}C^{p+q}},$$

$$E_{r}^{p,q}=\frac{Z_{r}^{p,q}}{B_{r}^{p,q}}.$$

The differential is given by

$$E_{r}^{p,q}\to \frac{Z_{r}^{p,q}}{Z_{r-1}^{p,q}}\xrightarrow{\sim} \frac{B_{r+1}^{p+r,q-r+1}}{B_{r}^{p+r,q-r+1}}\hookrightarrow E_{r}^{p+r,q-r+1},$$

then $E_{r}^{p,q}$ forms a spectral sequence. The objects $E_{0}^{p,q}=F^{p}C^{p+1}/F^{p+1}C^{p+q}$ and

$$E_{1}^{p,q}=h^{p+q}(gr^{p}C^{\bullet})\Rightarrow h^{p+q}(C^{\bullet}).$$

For a double complex we mean $(C^{\bullet,\bullet}, d_{h},d_{v})$ such that $d_{h}^{2}=d_{v}^{2}=0$ andn $d_{h}d_{v}+d_{v}d_{h}=0$. The total complex associated to $C^{\bullet,\bullet}$ is

$$Tot(C)^{n}=\bigoplus_{p+q=n}C^{p,q}$$

with differential $d=d_{h}+d_{v}$. Then we can define filtered complex by $F^{p}C^{p+q}=\bigoplus_{r\geq p}C^{p+q-r,r}$. Define the spectral sequence associated to $C^{\bullet, \bullet}$ with $E_{0}^{p,q}=C^{q,p}$, $d_{0}=d_{v}$, $E_{1}^{p,q}=H_{v}^{q}(C^{\bullet,p})$ and $d_{1}$ is induced by $d_{h}$, $E_{2}^{p,q}=H_{h}^{p}(H_{v}^{q}(C^{\bullet,\bullet}))$. Then

$$E_{2}^{p,q}=H_{h}^{p}(H_{v}^{q}(C^{\bullet,\bullet}))\Rightarrow H^{p+q}(Tot(C)).$$

The Cartan-Eilenberg resolution of $A^{\bullet}$ is a double complex $C^{\bullet,\bullet}$ such that $A^{p}\to C^{p,\bullet}$, $\mathop{\mathrm{ker}}(A^{p}\to A^{p+1})\to \mathop{\mathrm{ker}}(C^{p,\bullet}\to C^{p+1,\bullet})$, $\mathop{\mathrm{im}}(A^{p}\to A^{p+1})\to \mathop{\mathrm{im}}(C^{p,\bullet}\to C^{p+1,bullet})$ and $H^{p}(A^{\bullet})\to H_{h}^{p}(C^{\bullet, \bullet})$ give injective resolutions. Moreover, we sometimes require $0\to \mathop{\mathrm{ker}}d_{h}^{i,j}\to C^{i,j}\to \mathop{\mathrm{im}}d_{h}^{i,j}\to 0$ splits.

Every complex admits a Cartan-Eilenberg resolution. Using this, we can show

Theorem 1 (Grothendieck Spectral Sequence). Consider $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ which all all category of abelian sheaves on some projective variety. $F:\mathcal{A}\to \mathcal{B}$ and $G:\mathcal{B}\to \mathcal{C}$ are all additive functors such that $F$ maps injective elements into $G$-acyclic part. Then there is a spectral sequence $E_{r}^{p,q}$ such that

$$E_{2}^{p,q}=R^{p}G(R^{q}F(A))\Rightarrow R^{p+q}(F\circ G)(A).$$

Divisors, line bundles and linear systems

Definition 2 (Cartier Divisor). A Cartier divisor on $X$ is a global section of the sheaf $\mathcal{K}_{X}^{*} /\mathcal{O}_{X}^{*}$, where $\mathcal{K}_{X}$ of the total quotients. Denote the divisor group by

$$Div(X)=H^{0}(\mathcal{K}_{X}^{*}/\mathcal{O}_{X}^{*}).$$

We represent a Cartier divisor by $\{(U_{i},f_{i})\}$ where $f_{i}\in H^{0}(U_{i}, \mathcal{K}_{X}^{*})$ and satisfies the cocycle condition $f_{i}=g_{ij}f_{j}$, $g_{ij}\in H^{0}(U_{i}\cap U_{j}, \mathcal{O}_{X}^{*})$.

Remark 3. The support of divisor $D=\{(U_{i}, f_{i})\}$ is the set

$$\mathop{\mathrm{Supp}}(D)=\{x\in X|x_{i}\in U_{i} \text{ and } f_{i} \text{ is not unit in }\mathcal{O}_{X,x}\}.$$

If we regard $D$ as an Weil divisor $D=\sum a_{i}[D_{i}]$, the support $\mathop{\mathrm{Supp}}(D)=\coprod_{a_{i}\neq 0}\left| D_{i} \right|$.

Definition 4. A $k$-cycle on variety $X$ is a $\mathbb{Z}$-linear combination of irreducible subvariety of dimension $k$. We denote the group of all $k$-cycles by $Z_{k}(X)$.

For further definition of property of divisors, e.g. linear equivalence and effective divisors e.t.c. we refer to Hartshorne.

Example 5. For a morphism $f:Y\to X$ between schemes, define $f^{*}D$ by the pullback of all local section $\{f^{-1}U_{i}, f^{*}f_{i}\}$. To make $\{f^{-1}U_{i}, f^{*}f_{i}\}$ really define a Cartier divisor, the section must be nontrivial, which is true if we assume $\exists y\in Y$ such that $f(y)\notin Supp(D)$. (Note here we can assume $D$ effective and $f_{i}\in H^{0}(U_{i},\mathcal{O}_{U_{i}}^{*})$, and use linear extension to all divisors.) If $Y$ is reduced, the requirement is just that no components of $Y$ map into the support of $D$.

Definition 6 (Canonical Divisor). Let $X$ be a nonsingular complete variety of dimension $n$. Denote $\omega_{X}=\wedge^{n}\Omega_{X}$ to be the canonical bundle on $X$, and by $K_{X}$ we denote the canonical divisor. Then $\mathcal{O}(K_{X})=\omega_{X}$.

Definition 7. Let $\mathcal{L}$ be a line bundle, $V\subseteq H^{0}(X,\mathcal{L})$ is a finite dimensional subspace. Denote $\left| V \right| =\mathbb{P}(V)$ be the projective space of one dimensional subspace of $V$. We say $\left| V \right|$ is complete if $V=H^{0}(X,\mathcal{L})$.

Definition 8. Let $eval:V\otimes \mathcal{O}_{X}\to\mathcal{L}$ be the evaluation map. Let $\mathfrak{b}(\left| V \right| )$ be the image of $V\otimes \mathcal{L}^{*}\to \mathcal{O}_{X}$ given by the $eval$. The base point is the locus cutting down by the ideal $\mathfrak{b}(\left| V \right|)$, denoted by $Bs(\left| V \right|)$. We say $\mathcal{L}$ is base point free if $Bs(\left| V \right| )=\varnothing$.

Example 9. Assume $X$ is projective variety, $D$ is a divisor on $X$, then for $m,n\geq 1$, the natural homomorphism

$$H^{0}(\mathcal{O}(mD))\otimes H^{0}(\mathcal{O}(nD))\to H^{0}(\mathcal{O}((m+n)D))$$

defines inclusion $\mathfrak{b}(\left| mD \right| )\cdot \mathfrak{b}(\left| nD \right|) \subseteq \mathfrak{b}(\left| (m+n)D \right| )$.

Example 10. The linear system $\left| V \right|$ defines a morphsm

$$\varphi:X-Bs(\left| V \right| ) \to \mathbb{P}(\left| V \right| ),$$

defined by

$$\varphi:x\mapsto (s_{1}(x):\cdots: s_{n}(x)),$$

where $s_{1},\dots, s_{n}$ is a set of basis of $V$.

Note if we choose different basis of $V$, then the corresponding morphism will differ by an automorphism of $\mathbb{P}(V)$. If $\left| V \right|$ is base point free then $\varphi$ defines a morphism $X\to \mathbb{P}^{n}$.

Example 11. Suppose $W\subseteq V$ is a subspace, then $Bs(\left| V \right| )\subseteq Bs(\left| W \right| )$ and on $X-Bs(\left| W \right| )$, the morphism $\varphi_{W}=\pi\circ \varphi_{V}$, where $\pi:\mathbb{P}(V)-\mathbb{P}(V /W)\to \mathbb{P}(W)$ is the linear projection.

Definition 12. We say two Cartier divisors $D_{1}, D_{2}\in Div(X)$ are numberical equivalent if $D_{1}\cdot C=D_{2}\cdot C$ for every irreducible curve $C\subseteq X$. A divisor $D$ is called numerically trivial if it is numerical equivalent to zero.

Definition 13. Denote the set of all numerically trivial divisor by $Num(X)$. The Neron Severi group of $X$ is defined by $N^{1}(X)=Div(X) /Num(X)$.

Remark 14. This definition is slightly different with general definition, which is $Pic(X) /Pic^{0}(X)$. Using intersection theory on Picard scheme one can prove that they are the same with respect to $\mathbb{Z}, \mathbb{Q},\mathbb{R}$ coefficients.

Theorem 15. Neron Severi group is torsion free group of finite rank. Its rank is called Picard number and denoted by $\rho(X)$.

From above proof we also see the intersection number only depends on numerical equivalence, i.e. for $D_{1}^{i}\equiv_{num}D_{2}^{j}$, $D_{1}^{1}\cdots D_{1}^{\dim X-\dim V}\cdot V=D_{2}^{1}\cdots D_{2}^{\dim X-\dim V}\cdot V$.

Definition 16 (Rank of coherent sheaf). Let $X$ be variety of dimension $n$, $\mathcal{F}$ is coherent sheaf on $X$. The rank $\mathop{\mathrm{rk}}(\mathcal{F})$ is defined to be $length_{\mathcal{O}_{X,\xi}}\mathcal{F}_{\xi}$, where $\xi$ is the generic point of $X$.

Theorem 17 (Asymptotic Riemann Roch). Let $X$ be projective variety of dimension $n$. $D$ is a divisor on $X$, then $\chi(\mathcal{O}(mD))$ is a polynomial of degree $\leq n$.

$$\chi(\mathcal{O}(mD))=\frac{D^{n}}{n!}m^{n}+O(m^{n-1}).$$

More generally, for coherent sheaf $\mathcal{F}$ on $X$,

$$\chi(\mathcal{F}(mD))=\mathop{\mathrm{rk}}(\mathcal{F})\frac{D^{n}}{n!}m^{n}+O(m^{n-1}).$$

Proposition 18. Let $X$ be a projective variety of dimension $n$. Let $D$ be a divisor on $X$ with the property that
$h^{i}(\mathcal{O}(mD))=O(m^{n-1})$ for $i>0$. Fix a positive rational number $\alpha$ with $0<\alpha^{n}<D^{n}$. Then for $m\gg 0$, there exists for any smooth point $x\in X$ a divisor $E=E_{x}\in \left| mD \right|$ with $mult_{x}(D)>m\alpha$.

Amplitude

Definition 19. Let $X$ be a proper scheme, $\mathcal{L}$ is a line bundle on $X$,

1. $\mathcal{L}$ is very ample if the corresponding morphism $\varphi:X\to \mathbb{P}^{n}$ is closed immersion, and then $\mathcal{L}=\varphi^{*}\mathcal{O}_{\mathbb{P}^{n}}(1)$.

2. $\mathcal{L}$ is ample if $\mathcal{L}^{m}$ is very ample for some $m>0$.

Note this definition applys for proper scheme, therefore every immersion $X\to \mathbb{P}^{n}$ has closed image, and thus a closed immersion. The definition here is compactible with definition in Hartshorne.

Example 20. If $D_{1},\dots, D_{n}$ are ample divisors on dimension $n$ scheme $X$, then $m_{i}D_{i}$ are very ample for some $m_{i}>0$. Then

$$m_{1}D_{1}\cdots m_{n}D_{n}=m_{1}f^{*}H_{1}\cdots m_{n}f^{*}H_{n}=\deg \varphi \prod_{i=1}^{n}m_{i}H_{i}>0,$$

and thus $D_{1}\cdots D_{n}>0$.

Theorem 21. Let $\mathcal{L}$ be a line bundle on proper scheme $X$. Then the followings are equivalent:

1. $\mathcal{L}$ is ample;

2. For any coherent sheaf $\mathcal{F}$ on $X$, there exists positive number $m_{1}=m_{1}(\mathcal{F})$ such that $H^{i}(\mathcal{F}\otimes \mathcal{L}^{m})=0$ for all $m\geq m_{1}$;

3. For any coherent sheaf $\mathcal{F}$ on $X$, there exists positive number $m_{2}=m_{2}(\mathcal{F})$ such that $\mathcal{F}\otimes \mathcal{L}^{m}$ is generated by global sections for all $m\geq m_{2}$;

There exists a positive integer $m_{3}>0$ such that $\mathcal{L}^{m}$ is very ample for all $m\geq m_{3}$.

Example 22. For $\mathcal{L}$ a globally generated line bundle on a proper scheme $X$, the set

$$U=\{y\in X|\mathcal{L}\otimes m_{y}\text{ is globally generated}\}$$

is open.

To see the assertion, let

$$V=\{y\in X|H^{0}(\mathcal{L})\to \mathcal{L}\otimes \mathcal{O}_{X}/m_{y}^{2}\text{ is surjective}\}.$$

We first check $U=V$. Since $k(y)=\mathbb{C}$, $H^{0}(\mathcal{L}\otimes m_{y})\to \mathcal{L}\otimes m_{y}$ is surjective if and only if $H^{0}(\mathcal{L})\to \mathcal{L}_{y}\otimes m_{y}/m_{y}^{2}$ is surjective by Nakayama lemma. Also consider the diagram:

first row is exact since $\mathcal{L}$ is generated by global sections. $\alpha$ is surjective if and only if $\beta$ is. So $U=V$.

Next we construct coherent sheaf $\mathcal{P}$ on $X$ whose fibre at $y$ is $\mathcal{L}\otimes \mathcal{O}_{X}/m_{y}^{2}$ and a map $u:H^{0}(\mathcal{L})\to \mathcal{P}$ that on each fibre given by evaluations. If such $\mathcal{P}$ and $u$ exist, since $\mathop{\mathrm{coker}}(u)$ is coherent, the set $V=X-\mathop{\mathrm{Supp}}(\mathop{\mathrm{coker}}(u))$ is open. For such $\mathcal{P}$, consider the natural projections $X\xleftarrow{p}X\times X\xrightarrow{q}X$, take

$$\mathcal{P}=p_{*}(q^{*}\mathcal{L}\otimes (\mathcal{O}_{X\times X}/\mathcal{I}_{\Delta}^{2})),$$

where $\Delta$ is the diagonal. It’s easy to verify $\mathcal{P}$ has desired properties.

Example 23. If $D$ is ample divisor on $X$ and $E$ is any divisor, then $mD+E$ is very ample for $m\gg 0$. This is an easy corollary of the fact that tensor product of globally generated sheaves is still globally generated.

Example 24. If $\mathcal{L}$ and $\mathcal{M}$ are ample line bundles on projective schemes $X$ and $Y$ respectively, using Segre embedding, $\mathcal{L} \boxtimes \mathcal{M}$ is ample on $X\times Y$.

Proposition 25. Let $f:Y\to X$ be a finite morphism of proper schemes, and $\mathcal{L}$ an ample line bundle on $X$. Then $f^{*}\mathcal{L}$ is an ample line bundle on $Y$. In particular, if $Y\subseteq X$ is a subscheme of $X$, then the restriction $\mathcal{L}|_{Y}$ is ample.

Corollary 26. Suppose $\mathcal{L}$ is generated by global sections, let $\phi: X\to \mathbb{P}(H^{0}(\mathcal{L}))$ be the morphism corresponding to $\mathcal{L}$. Then the followings are equivalent:

1. $\mathcal{L}$ is ample.

2. $\phi$ is finite.

3. For every irreducible curve $C\subseteq X$, $\int_{C}c_{1}(\mathcal{L})>0$.

Proposition 27. Let $X$ be proper scheme and $\mathcal{L}$ is a line bundle on $X$,

1. $\mathcal{L}$ is ample if and only if $\mathcal{L}\otimes \mathcal{O}_{X_{red}}$ is ample on $X_{red}$.

2. $\mathcal{L}$ is ample if and only if $\mathcal{L}$ restricts to every irreducible component is ample.

Theorem 28 (Amplitude families). Let $f:X\to T$ be a proper morphism of schemes, $\mathcal{L}$ is a line bundle on $X$ and for $t\in T$, $\mathcal{L}_{t}=\mathcal{L}|_{X_{t}}$. If $\mathcal{L}_{0}$ is ample on $X_{0}$ for some point $0\in T$, then there exists an open neighborhood $U$ of $0$ such that $\mathcal{L}_{t}$ is ample for every $t\in U$.

Theorem 29 (Asymptotic Riemann Roch). Same set up as Theorem 17, but assume $D$ is ample. Then

$$h^{0}(\mathcal{O}(mD))=\frac{D^{n}}{n!}m^{n}+O(m^{n-1}).$$

More generally, for coherent sheaf $\mathcal{F}$ on $X$,

$$h^{0}(\mathcal{F}(mD))=\mathop{\mathrm{rk}}(\mathcal{F})\frac{D^{n}}{n!}m^{n}+O(m^{n-1}).$$

Example 30. Let $X$ be a projective scheme and $D$, $E$ ample divisors on $X$. Then there is an integer $m=m(D,E)$ such that

$$H^{0}(\mathcal{O}_{X}(aD))\otimes H^{0}(\mathcal{O}_{X}(bE))\to H^{0}(\mathcal{O}_{X}(aD+bE))$$

is surjective for $a,b\geq m$.

To see the assertion, consider the diagram $X\xleftarrow{p}X\times X\xrightarrow{q} X$ and the exact sequence

$$0\to \mathcal{I}_{\Delta}\to \mathcal{O}_{X\times X}\to \mathcal{O}_{\Delta}\to 0.$$

We write $(aD,bE)$ for the divisor $p^{*}aD+q^{*}bE$. By Kunneth formula, it suffices to show $H^{1}(X\times X, \mathcal{I}_{\Delta}(aD,bE))=0$ for all $a,b\geq m_{0}$. Pick a resolution of $\mathcal{I}_{\Delta}$ by

$$\cdots \to \oplus\mathcal{O}_{X\times X}(-p_{1}D,-p_{1}E)\to \oplus \mathcal{O}_{X\times X}(-p_{0}D,-p_{0}E)\to \mathcal{I}_{\Delta}\to 0,$$

we only need to show $H^{i}(X\times X,\mathcal{O}_{X\times X}(a-p_{i-1}D, b-p_{i-1}E))=0$ for $a,b\geq m$. But there are only finitely many nonzero terms, we can pick $m$ large enough to fulfill the requirement.

Remark 31. Note in previous proof, we use a subtle fact:

Let $\mathcal{F}_{\bullet}\to \mathcal{M}\to 0$ be a resolution of sheaves on dimension $n$ scheme $X$, and $H^{k+i}(\mathcal{F}_{i})=0$ for all $i$. Then $H^{k}(\mathcal{M})=0$.

The proof is simply split the resolution into short exact sequences and embed $H^{k}(\mathcal{M})$ into $H^{k+i+1}(\mathop{\mathrm{im}}(\mathcal{F}_{i+1}\to \mathcal{F}_{i}))$.

Definition 32 (Relative Amplitude). Let $f:X\to T$ be a proper morphism of schemes, let $\mathcal{L}$ be a line bundle on $X$,

1. $\mathcal{L}$ is very ample relative to $f$ if the canonical map $\rho: f^{*}f_{*}\mathcal{L}\to \mathcal{L}$ is surjective and defines an immersion

2. $\mathcal{L}$ is ample relative to $f$ if $\mathcal{L}^{m}$ is very ample for some $m>0$.

Clearly relative ampleness is local on the target.

Corollary 33. Let $f:X\to T$ be a proper morphism of schemes, and $\mathcal{L}$ a line bundle on $X$. Then the following are equivalent:

1. $\mathcal{L}$ is $f$-ample.

2. Given any coherent sheaf $\mathcal{F}$ on $X$, there exists a positive integer $m_{1} = m_{1}(\mathcal{F})$ such that

$$R^{i}f_{*}(\mathcal{F}\otimes\mathcal{L}^{m})=0 \text{ for all }i>0, m\geq m_{1}.$$

3. Given any coherent sheaf $\mathcal{F}$ on $X$, there is a positive integer $m_{2} = m_{2}(\mathcal{F})$ such that the canonical mapping

$$f^{*}f_{*}(\mathcal{F} \otimes\mathcal{L}^{m})\to \mathcal{F}\otimes\mathcal{L}^{m}$$

is surjective whenever $m \geq m_{2}$.

4. There is a positive integer $m_{3} > 0$ such that $\mathcal{L}^{m}$ is $f$-very ample for every $m \geq m_{3}$.

Example 34. Very ample relative to $f$ is equivalent to the existence of coherent sheaf $\mathcal{F}$ and an immersion $i:X\to \mathbb{P}(\mathcal{F})$ such that $\mathcal{L}=i^{*}\mathcal{O}_{\mathbb{P}(\mathcal{F})}(1)$.

One direction is clear, just take $\mathcal{F}=f_{*}\mathcal{L}$. For the converse, by the universal property, there is a surjection $f^{*}\mathcal{F}\to \mathcal{L}$, which factor through the pullback of $\sigma: \mathcal{F}\to f_{*}\mathcal{L}$ via $f$, i.e.

$$f^{*}\mathcal{F}\xrightarrow{f^{*}\sigma}f^{*}f_{*}\mathcal{L}\xrightarrow{\rho}\mathcal{L}.$$

Clearly $\rho$ is surjective and defines a morphism $j:X\to \mathbb{P}(f_{*}\mathcal{L})$ and therefore factors through $\mathbb{P}(f_{*}\mathcal{L})-\mathbb{P}(\mathop{\mathrm{coker}}\sigma)\to \mathbb{P}(\mathcal{F})$ and $j$. The image of $j$ lies in the open set $\mathbb{P}(f_{*}\mathcal{L})-\mathbb{P}(\mathop{\mathrm{coker}}\sigma)$. Since the projection $\mathbb{P}(f_{*}\mathcal{L})-\mathbb{P}(\mathop{\mathrm{coker}}\sigma)\to \mathbb{P}(\mathcal{F})$ is locally of finite type, we get $j$ is also an immersion.

Example 35. Suppose we have a diagram

$\mu$ is finite $T$-morphism. If $\mathcal{L}$ is $f$-ample, then $\mu^{*}\mathcal{L}$ is $g$-ample.

Note that for any coherent sheaf $\mathcal{F}$ on $Y$, $R^{i}\mu_{*}\mathcal{F}=0$, so by Grothendieck spectral sequence

$$E_{2}^{i,j}=R^{i}f_{*}(R^{j}\mu_{*}(\mathcal{F}\otimes \mu^{*}\mathcal{L}^{m}))\Rightarrow R^{i+j}g_{*}(\mathcal{F}\otimes \mu^{*}\mathcal{L}^{m})$$

vanishes for $i+j>0$.

Theorem 36. Let $f:X\to T$ be a proper morphism of schemes. $\mathcal{L}$ is a line bundle on $X$. Then $\mathcal{L}$ is $f$-ample if and only if $\mathcal{L}_{t}$ is ample on $X_{t}$ for every $t\in T$.

Theorem 37 (Nakai-Moishezon-Kleiman criterion). Let $\mathcal{L}$ be a line bundle on a projective scheme $X$. Then $\mathcal{L}$ is ample if and only if $$\int_{V} c_{1}(L)^{\dim(V)} > 0$$ for every positive-dimensional irreducible subvariety $V\subseteq X$ (including the irreducible components of $X$).