Ample and very ample divisors (Tag D-DIVAMPLE)
Let \(X\) be a noetherian scheme over a field \(k\), let \(D\) be a Cartier divisor, and put \(\mcl=\OO_X(D)\). The divisor \(D\) is very ample over \(k\) if \(\mcl\) is very ample over \(k\): there is an immersion \(\iota:X\to\PP_k^n\) with \(\iota^*\OO(1)\cong\mcl\) [@Har10a, Chapter II, §7].
(Ample)
On a noetherian scheme \(X\), a Cartier divisor \(D\) is ample if its associated invertible sheaf \(\OO_X(D)\) is ample [@Har10a, Chapter II, §7].
If \(X\) is of finite type over \(k\), then \(D\) is ample if and only if \(nD\) is very ample over \(k\) for some \(n>0\) [@Har10a, Theorem II.7.6]. If \(X\) is a projective integral variety over an algebraically closed field, then \(D\) is very ample if and only if its complete linear system is base-point free and its associated morphism is a closed immersion. Equivalently, \(D\) is linearly equivalent to a hyperplane section under some projective embedding [@Har10a, Chapter II, §7].
(Tensor products and finite pullback)
On a noetherian scheme, the tensor product of an ample invertible sheaf with a globally generated invertible sheaf is ample, as is the tensor product of two ample invertible sheaves. If \(\mcl\) is ample and \(\mcm\) is any invertible sheaf, then \(\mcm\otimes\mcl^{\otimes n}\) is ample for all sufficiently large \(n\) [@Har10a, Exercise II.7.5]. For a finite morphism \(f:Y\to X\) of noetherian schemes and an ample invertible sheaf \(\mcl\) on \(X\), the pullback \(f^*\mcl\) is ample. Indeed, for a coherent sheaf \(\mathcal F\) on \(Y\), the sheaf \(f_*\mathcal F\) is coherent by P-AGH255PUSHCOH. For large \(n\), pull back the global generators of \(f_*\mathcal F\otimes\mcl^{\otimes n}\) and use the surjection \(f^*f_*\mathcal F\to\mathcal F\) to generate \(\mathcal F\otimes(f^*\mcl)^{\otimes n}\). On \(f^{-1}(\Spec A)=\Spec B\), with \(\mathcal F\) represented by a finite \(B\)-module \(M\), this surjection is \(B\otimes_A M\to M\), \(b\otimes m\mapsto bm\).
(An arbitrary twist need not remain ample)
On \(\PP_k^1\), the sheaf \(\OO(1)\) is ample, but \(\OO(1)\otimes\OO(-2)\cong\OO(-1)\) is not. Every positive tensor power of \(\OO(-1)\) has no global sections, so it fails the ample global-generation condition even with the coherent sheaf \(\OO_{\PP^1}\) [@Har10a, Proposition II.5.13].
(Degree on an elliptic curve)
On a smooth projective genus-one curve over an algebraically closed field, an invertible sheaf of degree one is ample and has \(h^0=1\), so its complete linear system does not give an embedding. An invertible sheaf on this curve is very ample exactly when its degree is at least three; degree three embeds it as a plane cubic [@Har10a, Chapter IV, §§1 and 3].
(Hyperplane section)
For a projective integral variety \(X\subseteq\PP_k^n\) and a hyperplane \(H\) not containing \(X\), the hyperplane section \(X\cap H\) is the effective Cartier divisor cut out by the restricted linear form. Its associated invertible sheaf is \(\OO_X(1)\). Over an algebraically closed field, the hyperplane sections form the linear system \(\PP V\subseteq\abs{\OO_X(1)}\), where \(V\) is the image of \(H^0(\PP_k^n,\OO(1))\to H^0(X,\OO_X(1))\) [@Har10a, Chapter II, §7].
Serre's criterion for ampleness (Tag T-DIVSERRE)
Let \(X\) be a scheme of finite type over a Noetherian ring \(A\) and \(\mcl\) invertible. The following agree:
\(\mcl\) is ample;
for every coherent \(\mcf\) there is \(n_0\) with \(\mcf \tensor \mcl\tensorpower{}{n}\) globally generated for all \(n \geq n_0\).
If \(X\) is proper over \(A\), both are equivalent to: for every coherent \(\mcf\) there is \(n_0\) with \(H^i(X, \mcf \tensor \mcl\tensorpower{}{n}) = 0\) for all \(i > 0\) and all \(n \geq n_0\).
This criterion expresses ampleness as eventual global generation and, in the proper case, eventual vanishing of higher cohomology. Every argument that starts “twist by \(\OO(n)\) for \(n \gg 0\)” is an appeal to it.
On a projective scheme \(\OO(1)\) is ample, and the criterion is then Serre vanishing. The proper case is where the equivalence has content — it is how ampleness gets checked without exhibiting an embedding, and it is why ampleness is detectable from numerical data such as intersection numbers.
For \(X\) of finite type over a noetherian ring \(A\), an invertible sheaf \(\mcl\) is ample exactly when \(\mcl^{\otimes m}\) is very ample over \(A\) for some \(m>0\). When \(X\) is proper over \(A\), this holds exactly when \(H^i(X,\mcf\otimes\mcl^{\otimes n})=0\) for every coherent \(\mcf\), every \(i>0\), and every \(n\ge n_0(\mcf)\).
Ampleness of a divisor on a curve (Tag PR-DIVAMPCURVE)
Let \(D\) be a divisor on a smooth projective curve \(X\) of genus \(g\) over \(k = \bar{k}\).
\(D\) is ample if and only if \(\deg D > 0\).
\(\deg D \geq 2g\) implies \(\abs{D}\) is base-point free.
\(\deg D \geq 2g+1\) implies \(D\) is very ample.
On a curve ampleness collapses to a single integer, and this is the cleanest place to see the ample-against-very-ample distinction: positivity of the degree is ample, but very ampleness needs the degree to clear a genus-dependent threshold. Raising the degree past \(2g+1\) is exactly what taking powers achieves, which is the definition of ample in this case.
The thresholds are what Riemann–Roch gives: once \(\deg D > 2g-2\) the correction term vanishes and \(h^0\) is computable, so the conditions for base points and for separation become arithmetic. Both bounds are sharp on an elliptic curve, where \(g = 1\) and degree \(2\) gives the base-point-free double cover of \(\PP^1\) while degree \(3\) first embeds.
Coarser equivalences
Linear, algebraic, and numerical equivalence (Tag D-DIVEQUIV)
Let \(X\) be a smooth projective variety over an algebraically closed field, with intersection pairing \(D \cdot C = \deg \restrictionof{\OO_X(D)}{C}\) for \(C\) an irreducible curve. Let \(D_1,D_2\) be divisors on \(X\).
\(D_1\) and \(D_2\) are linearly equivalent, \(D_1 \sim D_2\), when \(D_1 - D_2 = \div(f)\) for a rational function \(f\).
\(D_1\) and \(D_2\) are algebraically equivalent, \(D_1 \sim_{\mathrm{alg}} D_2\), when they are related by the equivalence relation generated by the following: there are a nonsingular connected curve \(T\), points \(t_1,t_2\in T\), and a divisor \(\mathcal D\) on \(X\times T\), flat over \(T\), whose fibres over \(t_1\) and \(t_2\) are \(D_1\) and \(D_2\).
\(D_1\) and \(D_2\) are numerically equivalent, \(D_1 \equiv D_2\), when \((D_1 - D_2) \cdot C = 0\) for every irreducible curve \(C \subseteq X\).
The Néron–Severi group is \(\NS(X) = \Div(X)/{\sim_{\mathrm{alg}}} = \Pic(X)/\Pic^0(X)\), a finitely generated abelian group by the theorem of the base; its rank is the Picard number \(\rho(X)\). A divisor \(D\) is nef when \(D \cdot C \geq 0\) for every irreducible curve \(C\), and big when \(\limsup_{m\to\infty} h^0(\OO(mD))/m^{\dim X}>0\).
Linear equivalence implies algebraic equivalence, which implies numerical equivalence. The group \(\Pic(X)\) of linear equivalence classes can have a positive-dimensional component \(\Pic^0(X)\); the quotient \(\NS(X)\) is finitely generated, and the group of numerical classes is \(\NS(X)\) modulo its torsion subgroup.
Ampleness and nefness depend only on the numerical class: by the Nakai–Moishezon criterion, \(D\) is ample if and only if \(D^{\dim V}\cdot V>0\) for every closed subvariety \(V\) of positive dimension. In \(\NS(X)\otimes\RR\), the nef classes form the closure of the cone of ample classes (Kleiman), and a divisor is big if and only if some positive multiple is linearly equivalent to an ample divisor plus an effective divisor (Kodaira’s lemma).
On a smooth projective curve, algebraic and numerical equivalence both mean equality of degree, so \(\NS(X)=\ZZ\) by degree; linear equivalence is finer when the genus is positive, and nef means \(\deg\ge0\).
On a projective variety \(X\), whether a divisor is ample depends only on its numerical class, by the Nakai–Moishezon and Kleiman criteria; nefness depends only on it by definition. The group \(N^1(X)=\Pic(X)/{\equiv}\) is free abelian of rank \(\rho(X)\), and by Kleiman’s criterion the ample classes are the interior of the nef cone in \(N^1(X)\otimes\RR\).
Polarizations
Polarizations (Tag D-DIVPOLAR)
A polarization on a projective variety \(X\) is the class in \(\NS(X)\) of an ample line bundle. A polarized variety is a pair \((X, \mcl)\) with \(\mcl\) ample, considered up to isomorphisms preserving the class of \(\mcl\) in \(\NS(X)\). The degree of a polarization on \(X\) of dimension \(n\) is the intersection number \(\mcl^n\).
\((\PP^n, \OO(1))\) is polarized of degree \(1\), and a hypersurface of degree \(d\) with \(\OO(1)\) is polarized of degree \(d\). A K3 surface with an ample class \(h\) has a polarization of degree \(h^2 = 2g - 2\), and a quartic surface in \(\PP^3\) is a K3 surface of degree \(4\). For an abelian variety \(A\), a polarization is equivalently an isogeny \(A \to \dualof{A}\) of the form \(x \mapsto t_x^* \mcl \otimes \mcl^{-1}\); it is principal when this is an isomorphism, as for the theta divisor on the Jacobian of a curve.
Hyperplane sections
The linear system of hyperplane sections (Tag FE-DIVHYPSYS)
Let \(X \subseteq \PP^n\) be a projective variety over an algebraically closed field.
The hyperplane sections \(X \cap H\), for hyperplanes \(H \not\supseteq X\), form a base-point-free linear system of effective Cartier divisors on \(X\).
A general hyperplane section is smooth at every point where \(X\) is smooth. So it is smooth if \(X\) is smooth, and also if the singular locus of \(X\) is finite, since a general hyperplane misses finitely many points.
Normality of \(X\) does not make a general hyperplane section smooth. The cone \(X = V(x_0 x_1 - x_2^2) \subseteq \PP^4\) is a normal threefold singular along the line \(L = V(x_0, x_1, x_2)\). Every hyperplane meets \(L\), and \(X \cap H\) is singular at the point \(L \cap H\) whenever \(H \not\supseteq L\): in the plane \(\PP^3 \cong H\) it is again a quadric cone.