Effective adjunction theory

Abstract

Here we investigate the property of effectivity for adjoint divisors. Among others, we prove the following results: A projective variety X with at most canonical singularities is uniruled if and only if for each very ample Cartier divisor H on X we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$. Let X be a projective 4-fold, L an ample divisor and t an integer with $t \ge 3$. If $K_X+tL$ is pseudo-effective, then $H^0(X, K_X+tL) \ne 0$.

1 Introduction

Let X be a normal projective variety over the complex field $\mathbb {C}$; let $K_X$ be its canonical divisor. We assume that X has at most canonical singularities.

In the paper we fix a suitable Cartier divisor H on X and we discuss when the effectivity or non-effectivity of some adjoint divisors $aK_X + bH$ determines the geometry of X.

In the first part we consider the notion of Termination of Adjunction. This turns out to be rather delicate, since in the literature there are different meanings for such a property. The following are four possibilities, where $m_0$ and m are natural numbers.

• (A) For every (for some) big Cartier divisor H there exists $m_0=m_0(H)>0$ such that $mK_X+H \notin \overline{Eff(X)}$ (i.e. it is not pseudo-effective) for $m \ge m_0$.

• (B) For every big Cartier divisor H we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$.

• (C) For every very ample Cartier divisor H we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$.

• (D) For some (for every) big Cartier divisor $H_0$ we have $H^0(X, m_0K_X+kH_0)=0$ for every $k>0$ and some $m_0=m_0(k)>0$.

It is clear that (A) $\Longrightarrow$ (B) $\Longrightarrow$ (C) $\Longrightarrow$ (D).

We prove that these four definitions are equivalent and moreover that Adjunction Terminates in the above sense if and only if X is uniruled (see Theorem 3, Corollaries  1 and 2).

The results follow by some characterizations of pseudo-effective Cartier divisor (see Theorem 2), which are direct consequences of a fundamental result of Siu ([21]). The connection with uniruledness follows in turn from the fact that a projective variety X with canonical singularities is uniruled if and only if $K_X$ is not pseudo-effective (see [3], Corollary 0.3, or [5], Corollary 1.3.3).

A characterization of rationally connected manifolds along the same lines has been given in [6].

The examples described in [14], Theorem 39, show that, for varieties with singularities worst then canonical, uniruledness is not connected to Termination of Adjunction.

We consider also the following more general definition.

(C’):

Let H be an effective Cartier divisor on X. We say that Adjunction Terminates in the classical sense for H if there exists an integer $m_0 \ge 1$ such that

$$\begin{aligned} H^0(X, H+mK_X)=0 \end{aligned}$$

for every integer $m \ge m_0$.

We conjecture that such a definition is actually equivalent to the previous ones; a partial result in this direction is provided by Proposition 2. In dimension two, Castelnuovo and Enriques indeed proved that Condition (C’) implies that X is uniruled (see [7] and also [19]).

In the second part of the paper we assume that X is a projective variety of dimension n with at most terminal $\mathbb {Q}$-factorial singularities. We take a nef and and big Cartier divisor L on X and we call (X, L) a quasi polarized pair.

The following is a straightforward consequence of Theorem D in [5], see Remark 5 at the beginning of Sect. 5.

Proposition 1

Let (X, L) be a quasi polarized pair and $t >0$. If $K_X+tL\in \overline{Eff(X)}$, then there exists $N \in \mathbb {N}$ such that $H^0(X, N(K_X+tL)) \ne 0$.

Note that for $t=0$ the statement of the Proposition would amount to Abundance Conjecture, together with MMP.

The next Conjecture is an effective version of the above Proposition.

Conjecture 1

Let (X, L) be a quasi polarized pair and $t > 0$. If $K_X+tL \in \overline{Eff(X)}$, then $H^0(X, K_X+tL) \ne 0$.

The case $t=1$ is a version of the so-called Ambro–Ionescu–Kawamata conjecture, which is true for $n \le 3$ (see Theorem 1.5 in [13]), while for $t=n-1$ we recover a conjecture by Beltrametti and Sommese (see [4], Conjecture 7.2.7). Note that if Conjecture 1 holds for $t=1$ then it holds also for every $t > 0$.

In the paper we consider the following conjecture.

Conjecture 2

Let (X, L) be a quasi polarized pair and $s> 0$. Then $H^0(X, K_X+tL)=0$ for every integer t with $1 \le t \le s$ if and only if $K_X+sL$ is not pseudo-effective.

Since L is big, in particular pseudo-effective, then the if part is obvious. Note that Conjecture 2 for $s=1$ implies Conjecture 1.

We prove that Conjecture 2 is true for $s=n$ (see Proposition 4); we actually show that this case happens if and only if the pair (X, L) is birationally equivalent (via a 0-reduction, see the definition in the next section) to the pair $(\mathbb {P}^n, \mathcal {O}(1))$.

For $s=n-1$ the conjecture was essentially proved by Höring, see [13], Theorem 1.2. We prove a slightly more explicit version of his result (see Proposition 7), namely, we show that this case happens if and only if the pair (X, L) is birationally equivalent to a finite list of pairs.

Finally, we focus on the case $n = 4$ (see Theorem 8 and Proposition 6) and we generalize previous work by Fukuma ([11], Theorem 3.1).

2 Notation and preliminaries

Let X be a normal complex projective variety of dimension n. We adopt [15] and [16] as the standard references for our set-up. In particular, we denote by Div(X) the group of all Cartier divisors on X and by Num(X) the subgroup of numerically trivial divisors. The quotient group $N^1(X) = Div(X)/Num(X)$ is the Neron-Severi group of X.

In the vector space $N^1(X)_{\mathbb {R}} := N^1(X) \otimes \mathbb {R}$, whose dimension is $\rho (X) := rk N^1(X)$, we consider some convex cones.

• (a) $Amp(X) \subset N^1(X)_{\mathbb {R}}$ the convex cone of all ample $\mathbb {R}$-divisor classes; it is an open convex cone.

• (b) $Big(X) \subset N^1(X)_{\mathbb {R}}$ the convex cone of all big $\mathbb {R}$-divisor classes; it is an open convex cone.

• (e) $Eff(X) \subset N^1(X)_{\mathbb {R}}$ the convex cone spanned by the classes of all effective $\mathbb {R}$-divisors.

• (n) $Nef(X) = \overline{Amp(X)} \subset N^1(X)_{\mathbb {R}}$ the closed convex cone of all nef $\mathbb {R}$-divisor classes.

• (p) $\overline{Eff(X)} = \overline{Big(X)} \subset N^1(X)_{\mathbb {R}}$ the closed convex cone of all pseudo-effective $\mathbb {R}$-divisor classes.

The above definitions actually lean on some fundamental results like the openess of the ample and big cones, the facts that $int\{\overline{Eff(X)}\} = {Big(X)}$ and $Nef(X) = \overline{Amp(X)}$; for more details see [16].

Note that $Amp(X) \subset Nef(X) \cap Big(X)$ and that there are no inclusions between Nef(X) and Big(X).

Note also that if $\pi : X' \rightarrow X$ is a birational morphism and D is a Cartier divisor on X then D is big (resp. pseudo-effective) if and only if $\pi ^* D$ is big (resp. pseudo-effective).

We consider projective varieties with singularities of special type, as in the Minimal Model Program. For reader convenience we recall their definition (see [15], Definition 2.11 and Definition 2.12).

Definition 1

Let X be a normal projective variety. We say that X has canonical (respectively terminal) singularities if

• (i) $K_X$ is $\mathbb {Q}$-Cartier, and

• (ii) $\nu _*{\mathcal O}_{\tilde{X}}(m K_{\tilde{X}})= {\mathcal O}_{X}(m K_{X})$ for one (or for any) resolution of the singularities $\nu : {\tilde{X}} \rightarrow X$

(respectively

• ii) $\nu _*{\mathcal O}_{\tilde{X}}(m K_{\tilde{X}} - E)= {\mathcal O}_{X}(m K_{X})$ for one (or for any) resolution of the singularities $\nu : {\tilde{X}} \rightarrow X$, where $E \subset {\tilde{X}}$ is the reduced exceptional divisor).

In the category of projective varieties with canonical singularities the pseudo-effectivity of the canonical bundle is a birational invariant, as noticed by Mori in [18], (11.4.1). He actually conjectured the following beautiful result ([18], (11.4.2) and (11.5)), which was proved in [3], Corollary 0.3 and in [5], Corollary 1.3.3.

Theorem 1

Let X be a projective variety with at most canonical singularities. Then X is uniruled if and only if $K_X$ is not pseudo-effective.

As for the invariance of the global sections of adjoint bundles (or of pluri-canonical bundles if L is trivial) we have the following.

Lemma 1

Let $\pi : Y \rightarrow X$ be a birational morphism between projective varieties with at most canonical singularities, let L be a Cartier divisor on X and let $a, b \in \mathbb {N}$. Then

$$\begin{aligned} H^0(X, aK_X + b L) = H^0(Y, aK_Y + b \pi ^*(L)). \end{aligned}$$

Proof

Since Y and X have canonical singularities we have $\pi _* a K_Y = a K_X$. This is straightforward from the definition of canonical singularities and by taking a resolution of Y, $\nu : Y' \rightarrow Y$, and $\pi \circ \nu : Y' \rightarrow X$ as a resolution of X.

Since L is Cartier, by projection formula it follows

$$\begin{aligned} \pi _* (a K_Y + b \pi ^*(L))= \pi _* (a K_Y + \pi ^*(b L)) = \pi _*(a K_Y) + b L = a K_X + b L; \end{aligned}$$

by taking global sections we obtain our statement. $\square$

3 Termination of adjunction

Much of this section is based on the following Lemma, which was proved in the analytic setting by Siu (see [21], Proposition 1). For reader convenience we provide an algebraic proof relying on [17] (see also [20], Chapter V, Corollary 1.4).

Lemma 2

Let X be a smooth projective variety of dimension n and let H be a very ample divisor on X. If $G := (n+1)H+K_X$, then for every pseudo-effective divisor F on X we have $H^0(X, F+G) \ne 0$.

Proof

Since F is pseudo-effective we have that $F+H$ is big, hence there exists a positive integer $m >0$ such that $m(F+H) \sim A+E$ with A ample and E effective (see for instance [16], Corollary 2.2.7). Let $D := \frac{1}{m}E$ and $L := F+H$, so that $L-D = \frac{1}{m}A$ is big and nef; apply [17], Proposition 9.4.23, to get $H^0(X, K_X+L+kH + \mathcal {I}(D)) \ne 0$. Since the multiplier ideal $\mathcal {I}(D)$ is an ideal of $\mathcal {O}_X$, it follows that $H^0(X, K_X+L+kH) \ne 0$ for every $k \ge n$, i.e. $H^0(X, K_X+F+(k+1)H) \ne 0$ as soon as $k+1 \ge n+1$. $\square$

The following characterization of pseudo-effective divisors is probably well-known to the specialists; however, we did not find it explicitly in the literature.

Theorem 2

Let X be a smooth projective variety and let F be a divisor on X. The following statements, where m and N denote natural numbers, are equivalent:

• (i) $F\in \overline{Eff(X)}$ (i.e it is pseudo-effective).

• (ii) There is a big divisor G such that $H^0(X, N(mF+G)) \ne 0$ for every $m > 0$ and for some $N >0$.

• (iii) There is a big divisor G such that $H^0(X, mF+G) \ne 0$ for all $m > 0$.

• (iv) There is a very ample divisor G such that $H^0(X, mF+G) \ne 0$ for all $m > 0$.

• (v) For every big divisor H we have $H^0(X, mF+kH) \ne 0$ for all $m > 0$ and all $k \ge k_0(H)$.

Proof

First of all note that the implications (v) $\Longrightarrow$ (iv), (iv) $\Longrightarrow$ (iii) and (iii) $\Longrightarrow$ (ii) are obvious. Moreover (ii) $\Longrightarrow$ (i) follows from $F \equiv \lim _{m \rightarrow + \infty } \frac{mF+G}{m}$.

The difficult part is to prove (i) $\Longrightarrow$ (v); for this we use Lemma 2 together with Kodaira’s Lemma (see for instance [16], Proposition 2.2.6). Namely, let G be the divisor of Lemma 2; then $H^0(X, G) \ne 0$ (just take $F = {\mathcal O}_X$). If H is a big divisor on X, then by Kodaira’s Lemma $H^0(X, kH -G) \ne 0$ for every $k \ge k_0(H)$. Hence

$$\begin{aligned} \dim H^0(X, mF+kH)= & {} \dim H^0(X, mF+k_0H - G + G + (k-k_0)H)\ge \\\ge & {} \dim H^0(X, mF + (k-k_0)H +G) > 0, \end{aligned}$$

where the last inequality follows from Lemma 2 by taking as a pseudo-effective divisor $mF + (k-k_0)H$. $\square$

Remark 1

Note that (i) $\Longrightarrow$ (iii) is just Lemma 2, while (i) $\Longrightarrow$ (ii) follows easily from $int\{\overline{Eff(X)}\} = {Big(X)}$; this last fact was first noticed by Mori in [18], (11.3) on p. 318. Indeed, let $G \in Big(X)$ and $F \in \overline{Eff(X)}$; then the set $[G, F) := \{G +m F : m\in \mathbb {R}^+\}$ is contained in $int\{\overline{Eff(X)}\} =Big(X)$.

The next Theorem proves the equivalence of the different definitions of Termination of Adjunction stated in the Introduction.

Theorem 3

Let X be a projective variety with at most canonical singularities.

The following statements, where m and $m_0$ denote natural numbers, are equivalent:

• (i) X is uniruled (i.e. $K_X$ is not pseudo-effective).

• (ii) For every big Cartier divisor H there exists $m_0=m_0(H)>0$ such that $mK_X+H \notin \overline{Eff(X)}$ for $m \ge m_0$.

• (iii) For every big Cartier divisor H we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$.

• (iv) For every very ample Cartier divisor H we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$.

• (v) For some big Cartier divisor $H_0$ we have $H^0(X, m_0K_X+kH_0)=0$ for every $k>0$ and some $m_0=m_0(k)>0$.

Proof

(i) $\Longrightarrow$ (ii) is implied by the properties of the cone described in Sect. 2; indeed, it follows by contradiction from $K_X \equiv \lim _{m \rightarrow + \infty } \frac{mK_X+H}{m}$. (ii) $\Longrightarrow$ (iii), (iii) $\Longrightarrow$ (iv) and (iv) $\Longrightarrow$ (v) are straightforward. (v) $\Longrightarrow$ (i) requires a resolution of the singularities $\nu : {\tilde{X}} \rightarrow X$. Assume by contradiction that X is not uniruled. Therefore also ${\tilde{X}}$ is not uniruled and $K_{\tilde{X}}$ is pseudo-effective. If H is any big Cartier divisor on X, then ${\tilde{H}}= \nu ^*(H)$ is big and by [16], Corollary 2.2.7, we have $l{\tilde{H}}=A+N$ with A ample and N effective for some $l>0$. It follows that $hl{\tilde{H}}=hA+hN$ with hA very ample for some $h>0$. Hence, by Lemma 1, for every $m_0 >0$ we have $\dim H^0(X, m_0K_X+(n+1)hlH) = \dim H^0({\tilde{X}}, m_0K_{\tilde{X}}+(n+1)hl{\tilde{H}})= \dim H^0({\tilde{X}}, (m_0-1)K_{\tilde{X}}+(K_{\tilde{X}}+(n+1)hA)+(n+1)hN) \ge \dim H^0({\tilde{X}}, (m_0-1)K_{\tilde{X}}+(K_{\tilde{X}}+(n+1)hA))$. Lemma  2 says that this last term is positive, thus contradicting our assumption. $\square$

Remark 2

Note that Mori in [18], (11.4) on p. 318, suggests that in principle (i) could have been stronger then (iv): We say that X is $\kappa$-uniruled if $K_X$ is not pseudo-effective. We note that $\kappa$-uniruledness is slightly stronger than saying that adjunction terminates, i.e. $H^0(X, mK_X+H)=0$ for each very ample divisor H and some $m=m(H)>0$.

The following two corollaries show that the two formulations, respectively for some and for every, of (A) and (D) in the Introduction are equivalent.

Corollary 1

Let X be a projective variety with at most canonical singularities.

The following statements, where m and $m_0$ denote natural numbers, are equivalent:

• (i) For every big Cartier divisor H there exists $m_0=m_0(H)>0$ such that $mK_X+H \notin \overline{Eff(X)}$ for $m \ge m_0$.

• (ii) For some big Cartier divisor $H_0$ there exists $m_0=m_0(H_0)>0$ such that $mK_X+H_0 \notin \overline{Eff(X)}$ for $m \ge m_0$.

Proof

It is obvious that (i) implies (ii). Conversely, if (ii) holds then $K_X$ is not pseudo-effective, hence X is uniruled. It follows from Theorem 3 that (i) holds. $\square$

Corollary 2

Let X be a projective variety with at most canonical singularities.

The following statements, where m and $m_0$ denote natural numbers, are equivalent:

• (i) For some big Cartier divisor $H_0$ we have $H^0(X, m_0K_X+kH_0)=0$ for every $k>0$ and some $m_0=m_0(k)>0$.

• (ii) For every big Cartier divisor H we have $H^0(X, m_0K_X+kH)=0$ for every $k>0$ and some $m_0=m_0(k,H)>0$.

Proof

It is obvious that (ii) implies (i). Conversely, if (i) holds then by Theorem 3 X is uniruled, i.e. $K_X$ is not pseudo-effective. Assume by contradiction that there exist a big divisor H and some $k_0>0$ such that $H^0(X, mK_X+k_0H)\ne 0$ for every $m >0$. Then $K_X = \lim _{m \rightarrow + \infty } \frac{mK_X+k_0H}{m}$ is pseudo-effective, a contradiction. $\square$

As pointed out by the referee, since every divisor is a difference of very ample ones, (C) is actually equivalent to the following stronger condition.

(C*):

For every Cartier divisor D we have $H^0(X, m_0K_X+D)=0$ for some $m_0=m_0(D)>0$.

The following is a more general definition of Termination of Adjunction.

Definition 2

(Condition (C’)) Let X be a normal projective variety; let H be an effective Cartier divisor on X. We say that Adjunction Terminates in the classical sense for H if there exists an integer $m_0 \ge 1$ such that

$$\begin{aligned} H^0(X, H+mK_X)=0 \end{aligned}$$

for every integer $m \ge m_0$.

We conjecture that such a definition is actually equivalent to the previous ones. The following partial result in this direction is straightforward.

Proposition 2

Let X be a projective variety with canonical singularities. Let H be any effective divisor and assume that Adjunction Terminates in the classical sense for H. Then X has negative Kodaira dimension.

Proof

Recall that the Kodaira dimension of a singular variety is defined to be the Kodaira dimension of any smooth model (see for instance [16], Example 2.1.5). Assume by contradiction that X has non-negative Kodaira dimension, i.e. $H^0({\tilde{X}}, n_0 K_{\tilde{X}}) \ne 0$ for some integer $n_0 \ge 1$, where $\nu : {\tilde{X}} \rightarrow X$ is any resolution of the singularities. Since X has canonical singularities, from Lemma 1 it follows that $H^0(X, n_0 K_X) = H^0({\tilde{X}}, n_0 K_{\tilde{X}}) \ne 0$. Hence $H^0(X, H+ n n_0 K_X) \ne 0$ for every integer $n \ge 1$, contradicting the assumption that $H^0(X, H+mK_X)=0$ for $m>> 0$. $\square$

Together with the standard conjecture that negative Kodaira dimension implies uniruledness (see for instance [18], (11.5) on p. 319, and [3], Conjecture 0.1), from Proposition  2 it would follow that Termination of Adjunction in the classical sense implies uniruledness. In dimension two such an implication holds unconditionally, as it was proved by Castelnuovo and Enriques in [7] (for a modern proof we refer to [19]).

We conclude this section with a characterization of uniruled varieties which may suggest a different way to consider (effective) termination of adjunction. It follows as a straightforward consequence of Lemma 2 and the main result in [3].

Proposition 3

Let X be a smooth projective variety of dimension n and let H be a very ample divisor on X. If $H^0(X, mK_X+(n+1)H)=0$ for some natural number $m \ge 1$, then X is uniruled.

Proof

Assume by contradiction that X is not uniruled, so that $K_X$ is pseudo-effective by [3]. Lemma 2 with $F = (m-1)K_X$ gives the sought-for contradiction. $\square$

Theorem 3.1 in [8] gives a statement similar to the last proposition; there the variety is singular and H is just nef and big. However $m >1$ and H has to be multiplied by a higher number, for instance $n^2$.

4 Quasi polarized pairs

A quasi polarized pair is a pair (X, L) where X is a projective variety with at most $\mathbb {Q}$-factorial terminal singularities and L is a nef and big Cartier divisor on X. If L is ample we call the pair (X, L) a polarized pair.

In [1], Sect. 4, following T. Fujita’s ideas as revisited by A. Höring in [13] and using the MMP developed in [5], we described a MMP with scaling related to divisors of type $K_X + rL$, for r a positive rational number.

In particular we introduced the 0-reduction of a quasi polarized pair (X, L) (see [1], Definition 4.4) as quasi polarized pair $(X',L')$ birational to (X, L) obtained from (X, L) via a Minimal Model Program with scaling:

$(X,L) \sim (X, \Delta ) :=(X_0, \Delta _0) \rightarrow ----\rightarrow (X_s, \Delta _s) \sim (X',L'),$ which contracts or flips all extremal rays $\mathbb {R}^+[C]$ on X such that $L\cdot C =0$.

At every step of the MMP given above, we have a quasi polarized variety $(X_i, L_i)$ with at most terminal $\mathbb {Q}$-factorial singularities.

If $\pi _i:(X_{i}, \Delta _i) \rightarrow (X_{i+1}, \Delta _{i+1})$ is birational then $L_i = \pi _i^*(L_{i+1})$, while if $\pi _i:(X_i, \Delta _i) \rightarrow (X_{i+1}, \Delta _{i+1})$ is a flip then $L_{i}$ and $\pi _i^*(L_{i+1})$ are isomorphic in codimension one.

Remark 3

By using Lemma 1 and Hartogs theorem we deduce

$$\begin{aligned} H^0(X, aK_{X} +bL) = H^0(X', aK_{X'} +bL') \end{aligned}$$

for $a, b\in \mathbb {N}.$

The following has been proved in [1], Theorem 5.1 and in [12], Proposition 1.3.

Theorem 4

Let (X, L) be a quasi polarized pair. Then $K_X + t L$ is pseudo-effective for all $t \ge n$ unless the 0-reduction $(X',L')$ is $(\mathbb {P}^n, {\mathcal O}(1))$. Actually, $K_X + (n-1)L$ is pseudo-effective unless $(X',L')$ is one of the following pairs:

• $(\mathbb {P}^n, {\mathcal O}(1))$,

• $(Q, {\mathcal O}(1)_{|Q})$, where $Q\subset \mathbb {P}^{n+1}$ is a quadric,

• $C_n(\mathbb {P}^2, {\mathcal O}(2))$, a generalized cone over $(\mathbb {P}^2, {\mathcal O}(2))$,

• X has the structure of a $\mathbb {P}^{n-1}$-bundle over a smooth curve C and L restricted to any fiber is ${\mathcal O}(1)$.

Moreover, except in the above cases, $K_{X'} + (n-1)L'$ is nef.

The first-reduction of a quasi polarized pair (X, L) (see [1], Definition 5.5) is a quasi polarized pair $(X'',L'')$ birational to (X, L) obtained from a 0-reduction $(X',L')$ via a morphism $\rho : X' \rightarrow X''$ consisting of a series of divisorial contractions to smooth points, which are weighted blow-ups of weights $(1,1,b, \ldots , b)$ with $b \ge 1$ (see [2], Theorem 1.1).

Remark 4

According to [1], Proposition 5.4, we have

$$\begin{aligned} H^0(K_{X} +tL) = H^0(K_{X''} +tL'') \end{aligned}$$

for any $0\le t \le n-2$.

The following has been proved in [1], Theorem 5.7.

Theorem 5

Let (X, L) be a quasi polarized pair.

$K_{X} + (n-2)L$ is not pseudo-effective if and only if any first-reduction $(X'', L'')$ is either one of the pairs listed in the statement of Theorem 4 or one of the following pairs:

• a del Pezzo variety, that is $-K_{X''} \sim _\mathbb {Q}(n-1)L$ with L ample,

• $(\mathbb {P}^4, {\mathcal O}(2))$,

• $(\mathbb {P}^3, {\mathcal O}(3))$,

• $(Q,{\mathcal O}(2)_{|Q})$, where $Q\subset \mathbb {P}^{4}$ is a quadric,

• X has the structure of a quadric fibration over a smooth curve C and L restricted to any fiber is ${\mathcal O}(1)_{|Q}$,

• X has the structure of a $\mathbb {P}^{n-2}$-bundle over a normal surface S and L restricted to any fiber is ${\mathcal O}(1)$,

• $n=3$, X is fibered over a smooth curve Z with general fiber $\mathbb {P}^{2}$ and L restricted to it is ${\mathcal O}(2)$.

If $K_{X} + (n-2)L$ is pseudo-effective then on any first-reduction $(X'', L'')$ the divisor $K_{X''} + (n-2)L''$ is nef.

The following definition was given by Höring (see ([13], Definition 1.2).

Definition 3

A quasi polarized pair (X, L) is a (generalized) scroll if X is smooth and there is a fibration $X \rightarrow Y$ onto a projective manifold Y such that the general fiber F admits a birational morphism $\tau : F \rightarrow \mathbb {P}^m$ and that $\mathcal {O}_F(L) = \tau ^*\mathcal {O}_{\mathbb {P}^m}(1)$. A quasi polarized pair (X, L) is birationally a scroll if there is a birational morphism $\nu : X' \rightarrow X$ such that $(X', \nu ^* L)$ is a (generalized) scroll.

The next is Theorem 1.4 in [13].

Theorem 6

Let (X, L) be a quasi polarized pair. If (X, L) is not birationally a scroll then $\Omega _X \otimes L$ is generically nef.

A key step in the proofs of Theorem 7 and of Theorem  8 is the following lemma due to Höring (see [13], p. 741, Step 2 in the proof of Theorem 1.2).

Lemma 3

Let (X, L) be a quasi polarized pair. Assume that $K_X+(n-2)L$ is pseudo-effective and that $K_X+(n-1)L$ is nef and big. Then

$$\begin{aligned} L^{n-2} [(2(K_X^2 +c_2 (X))+6nL K_X +(n+1)(3n-2)L^2] > 0. \end{aligned}$$

We consider now a quasi polarized pair (X, L) and we assume moreover that X is smooth. We borrow from Y. Fukuma the following set-up for the computation of the Hilbert polynomial of $K_X+tL$.

Let

$$\begin{aligned} F_{0}(t):= & {} \dim H^0(X, K_X +tL),\\ F_{i}(t):= & {} F_{i-1}(t+1) - F_{i -1}(t) \hbox { for every integer } i \hbox { with } 1\le i\le n. \end{aligned}$$

The following statement can be easily checked by reverse induction on $b \le a$.

Lemma 4

Fix an integer $a \ge 1$. If $F_{0}(t)=0$ for every integer t with $1 \le t \le a$, then $F_{a-b}(c)=0$ for all integers b, c with $1 \le c \le b \le a$.

If one defines

$$\begin{aligned} A_i(X,L) := F_{n-i}(1) \end{aligned}$$

then it follows easily that

$$\begin{aligned} \dim H^0(X, K_X+tL) = \sum _{j=0}^n \left( {\begin{array}{c}t-1\\ n-j\end{array}}\right) A_j(X,L). \end{aligned}$$

(1)

Moreover, by taking $a = n-i+1$ and $b=c=1$ in Lemma  4, we obtain the following implication.

Corollary 3

If $H^0(X, K_X+tL)=0$ for every integer t with $1 \le t \le n-i+1$, then $A_i(X,L)=0$.

On the other hand, by Kawamata–Viehweg vanishing theorem and Serre duality, we have $\dim H^0(X, K_X+tL) = \chi (X,-tL)$; therefore from the Riemann–Roch theorem we obtain the following explicit computations (for further details, see [9], (2.2), and [10], Proposition 3.2).

Lemma 5

Let (X, L) be a polarized manifold of dimension n and let g(X, L) denote the sectional genus of (X, L). Then we have

$$\begin{aligned} A_0(X,L)= & {} L^n \\ A_1(X,L)= & {} g(X,L)+L^n-1 \\ 24\cdot A_2(X,L)= & {} L^{n-2} [(2(K_X^2+c_2(X))+6nL K_X +(n+1)(3n-2)L^2] \\ 48\cdot A_3(X,L)= & {} (n-2)(n^2-1)L^n+ n(3n-5)K_X L^{n-1} + \\&+ 2(n-1) K_X^2 L^{n-2} + 2c_2(X)(K_X+(n-1)L) L^{n-3}. \end{aligned}$$

5 Polarized abundance

The aim of this section is to argue around the Conjectures stated in the introduction.

We start showing that Proposition 1 is a direct consequence of (the more general) Theorem D in [5].

Remark 5

Let (X, L) be a quasi-polarized variety and let t be a positive rational number. Then there exists an effective $\mathbb {Q}$-divisor $\Delta ^t$ on X such that $\Delta ^t \sim _{\mathbb {Q}} t L$ and $(X, \Delta ^t)$ is Kawamata log terminal. This is well-known to the specialists, a proof can be found in [1]. If $K_X+tL\in \overline{Eff(X)}$, then $K_X+\Delta ^t \in \overline{Eff(X)}$ and by [5], Theorem D, there exists an $\mathbb {R}$-divisor $D \ge 0$ such that $K_X+\Delta ^t \sim _{\mathbb {R}} D$. That is, there exists $N \in \mathbb {N}$ such that $H^0(X, N(K_X+tL))>0$.

We consider Conjecture 2; for $s=n$ we recover the following easy fact.

Proposition 4

Let (X, L) be a quasi polarized pair of dimension n. We have $H^0(X, K_X+tL)=0$ for every integer t with $1 \le t \le n$ if and only if $K_X+ nL$ is not pseudo-effective. Moreover this is the case if and only if the 0-reduction $(X',L')$ of the pair (X, L) is $(\mathbb {P}^n, \mathcal {O}(1))$.

Proof

By Remark 3 we have $H^0(X, K_{X} +tL) = H^0(X', K_{X'} +tL')$ for any $t\ge 0$. Hence if $H^0(X, K_X+tL)=0$ for every integer t with $1 \le t \le n$ then from Corollary 3 it follows that $A_1(X',L')=g(X',L')+L'^n-1=0$. Since we have $g(X',L')=0$ and $L'^n=1$ if and only if $(X',L') = (\mathbb {P}^n, \mathcal {O}(1))$, the claim follows from [1], Theorem 5.1 (2). $\square$

Next, for $s=n-1$, the following is a slightly more explicit version of [13], Theorem 1.2; the proof is essentially the one of [13].

Theorem 7

Let (X, L) be a quasi polarized pair of dimension n. We have $H^0(X, K_X+tL)=0$ for every integer t with $1 \le t \le n-1$ if and only if $K_X+(n-1)L$ is not pseudo-effective.

That is, by Theorem 4, if and only if the 0-reduction $(X',L')$ of the pair (X, L) is one of the following:

• (i) $(\mathbb {P}^n, {\mathcal O}(1))$,

• (ii) $(Q, {\mathcal O}(1)_{|Q})$, where $Q\subset \mathbb {P}^{n+1}$ is a quadric,

• (iii) $C_n(\mathbb {P}^2, {\mathcal O}(2))$, a generalized cone over $(\mathbb {P}^2, {\mathcal O}(2))$,

• (iv) X has the structure of a $\mathbb {P}^{n-1}$-bundle over a smooth curve C and L restricted to any fiber F is ${\mathcal O}(1)$.

Proof

Let $(X',L')$ be the 0-reduction of the pair (X, L) and let $({\tilde{X'}},\tilde{L}')$ be its desingularization (namely, $\nu : {\tilde{X'}} \rightarrow X'$ and ${\tilde{L'}}= \nu ^*(L'))$.

By Remark 3 and Lemma 1 we have

$$\begin{aligned} H^0(X, K_{X} +tL) = H^0(X', K_{X'} +tL')= H^0({\tilde{X}}', K_{\tilde{X'}} +t{\tilde{L'}}) \end{aligned}$$

for any $t\ge 0$.

The if part is obvious. In order to prove the only if part, assume that $H^0(X, K_{X} +tL) = H^0(X', K_{X'}$ $+tL')= H^0({\tilde{X'}}, K_{\tilde{X'}} +t{\tilde{L'}}) =0$ for every integer t with $1 \le t \le n-1$. Corollary 3 implies that

$$\begin{aligned} A_2({\tilde{X'}}, \tilde{L}')=0. \end{aligned}$$

(2)

Assume by contradiction that $(X',L')$ is not one of the pairs in (i), (ii), (iii), (iv); then, by Theorem 4, $K_{X'}+(n-1)L'$ is nef. The required contradiction is provided by [13], Theorem 1.2. $\square$

The next step $s=n-2$ should work as follows.

Conjecture 3

Let (X, L) be a quasi polarized manifold of dimension n. We have $H^0(X, K_X+tL)=0$ for every integer t with $1 \le t \le n-2$ if and only if $K_X+(n-2)L$ is not pseudo-effective, that is if and only if the first-reduction $(X'',L'')$ is one of the pairs (X, L) listed in Theorems 4 and 5.

Once again, the if part is obvious. Conversely, from Corollary 3 it follows that $A_3(X,L)=0$, but the proof of the only if part seems to be elusive.

From now on, we focus on the case $n=4$; here formula ( 1) reads simply as:

$$\begin{aligned} H^0(X, K_X+tL)= & {} \left( {\begin{array}{c}t-1\\ 4\end{array}}\right) A_0(X,L) + \left( {\begin{array}{c}t-1\\ 3\end{array}}\right) A_1(X,L) \nonumber \\&+\left( {\begin{array}{c}t-1\\ 2\end{array}}\right) A_2(X,L) +\left( {\begin{array}{c}t-1\\ 1\end{array}}\right) A_3(X,L) + \left( {\begin{array}{c}t-1\\ 0\end{array}}\right) A_4(X,L) \nonumber \\ \end{aligned}$$

(3)

where

$$\begin{aligned} A_1(X,L)= & {} g(X,L) + L^4 - 1, \\ A_2(X,L)= & {} \dim H^0(X, K_X+3L)-2\dim H^0(X, K_X+2L) \\&+ \dim H^0(X, K_X+L),\\ A_3(X,L)= & {} \dim H^0(X, K_X+2L)-\dim H^0(X, K_X+L), \\ A_4(X,L)= & {} \dim H^0(X, K_X+L). \end{aligned}$$

We prove the following generalization of [11], Theorem 3.1.

Theorem 8

Let (X, L) be a polarized manifold of dimension 4 and let t be an integer with $t \ge 3$. If $K_X+tL$ is pseudo-effective, then $H^0(X, K_X+tL) \ne 0$. In particular,

• $H^0(X, K_X+tL) \ne 0$ for $t\ge 5$

• $H^0(X, K_X+4L)=0$ if and only if (X, L) is $(\mathbb {P}^4, \mathcal {O}(1))$

• $H^0(X, K_X+3L)=0$ if and only if (X, L) is either $(Q, {\mathcal O}(1)_{|Q})$, where $Q\subset \mathbb {P}^{5}$ is a quadric, or X has the structure of a $\mathbb {P}^{3}$-bundle over a smooth curve C and L restricted to any fiber is ${\mathcal O}(1)$.

Proof

Since L is ample (X, L) is a 0-reduction, in particular by Theorem 4 we can assume that $K_X+tL$ is nef for $t \ge 4$. We can also assume that $K_X+3L$ is nef. Indeed, if not then (X, L) is one of the exceptions listed in the statement of Theorem 4. If (X, L) is $(\mathbb {P}^4, \mathcal {O}(1))$ or $(Q, \mathcal {O}(1))$, where $Q \subset \mathbb {P}^5$ is a quadric hypersurface, then Theorem 8 is obvious. The case of a generalized cone over $(\mathbb {P}^2, \mathcal {O}(2))$ does not occur since X is smooth, while the case of a $\mathbb {P}^3$-bundle over a smooth curve will be considered in Proposition 5.

Now, assume that $\Omega _X \otimes L$ is generically nef. By using the formulas in Lemma 5 and Miyaoka inequality as stated in [13], Corollary 2.11, with $D:=4L$, we compute:

$$\begin{aligned} A_2(X,L)\ge & {} \frac{1}{24}\left( 2(K_X+3L)^2 L^2 + 6 (K_X+3L) L^3 + 2L^4 \right) \\ A_3(X,L)\ge & {} - \frac{1}{24} (K_X+3L) L^3. \end{aligned}$$

Hence from ( 3) and the nefness of $K_X+3L$ it follows that

$$\begin{aligned} \dim H^0(X, K_X+tL) \ge (t-1) A_3(X,L) + \frac{(t-1)(t-2)}{2} A_2(X,L) > 0 \end{aligned}$$

for every $t \ge 3$.

Finally, assume that $\Omega _X \otimes L$ is not generically nef. By Theorem 6 and Lemma 1 we may assume that X is a (generalized) scroll and the claim is a consequence of the following proposition. $\square$

Proposition 5

Let (X, L) be a generalized scroll of dimension 4 and let t be an integer such that $t \ge 3$. If $K_X+tL$ is nef, then $H^0(X, K_X+tL) \ne 0$.

Proof

Let $X \rightarrow Y$ be the scroll fibration and let F be the generic fiber with a birational morphism $\tau : F \rightarrow \mathbb {P}^m$ as in Definition 3.

If $X = \mathbb {P}^4$ the claim is obvious; therefore we can assume that $m \le 3$ and that $A_1(X,L) = g(X,L) + L^4 - 1 > 0$ (since we have $g(X,L)=0$ and $L^4=1$ if and only if $(X,L) = (\mathbb {P}^4, \mathcal {O}(1))$). We also have that $A_0(X,L) = L^4 \ge 1$ and $A_4(X,L) = \dim H^0(X, K_X+L) \ge 0$.

If $m=3$, then $K_X+sL \vert _F = \tau ^*\mathcal {O}_{\mathbb {P}^3}(-4+s)$, hence $H^0(X,K_X+sL)=0$ for $s \le 3$. Thus we have $A_2(X,L)=A_3(X,L)=0$ and from ( 3) it follows that for $t \ge 4$ we have

$$\begin{aligned} \dim H^0(X, K_X+tL) \ge A_1(X,L) > 0 \end{aligned}$$

If $m = 2$, then $K_X+sL \vert _F = \tau ^*\mathcal {O}_{\mathbb {P}^2}(-3+s)$, hence $H^0(X,K_X+sL)=0$ for $s \le 2$. In particular, we have $A_3(X,L)=0$ and $A_2(X,L)=\dim H^0(X, K_X+3L)$.

For $t=3$, i.e. if we assume $K_X+3L$ is nef, by Theorem 1.2 in [13] we must have $H^0(X,K_X+3L) \ne 0$ since $H^0(X,K_X+sL)=0$ for $s \le 2$.

For $t \ge 4$ we deduce from ( 3) that

$$\begin{aligned} \dim H^0(X, K_X+tL) \ge A_1(X,L) > 0. \end{aligned}$$

If $m=1$, then $K_X+sL \vert _F = \tau ^*\mathcal {O}_{\mathbb {P}^1}(-2+s)$, hence $H^0(X,K_X+L)=0$. In particular, we have $A_3(X,L) \ge 0$.

If $H^0(X,K_X+2L)=0$, then $A_2(X,L)=\dim H^0(X, K_X+3L)$ and we conclude exactly as in the previous case $m = 2$.

If $H^0(X,K_X+2L) \ne 0$, then $K_X+2L$ is pseudo-effective and $K_X+3L$ is pseudo-effective and big.

Passing to the 0-reduction we may assume that $K_X+3L$ is nef and big. Therefore Lemma 3 applies and by Lemma 5 we get $A_2(X,L)>0$.

Hence from ( 3) it follows that for $t \ge 3$ we have

$$\begin{aligned} \dim H^0(X, K_X+tL) \ge A_2(X,L) > 0. \end{aligned}$$

$\square$

The statement of Theorem 8 should hold also for $t=2$, but we have only the following partial result.

Proposition 6

Let (X, L) be a polarized manifold of dimension 4. If $K_X+2L$ is pseudo-effective, then $H^0(X, K_X+2L) \ne 0$ unless $\Omega _X<\frac{1}{2}L>$ is not generically nef.

Proof

By Theorem 5 and Remark 4 we may assume that $K_X+2L$ is nef.

Assume that $\Omega _X<\frac{1}{2}L>$ is generically nef. By using the formula for $A_3(X,L)$ in Lemma 5 and Miyaoka inequality, as stated in [13], Corollary 2.11, with $D:=2L$, we compute:

$$\begin{aligned} A_3(X,L) \ge \frac{1}{16} (K_X+2L)^2L^2 + \frac{1}{12}(K_X+2L)L^3 + \frac{1}{48} L^4. \end{aligned}$$

Hence from ( 3) it follows that

$$\begin{aligned} \dim H^0(X, K_X+tL) \ge A_3(X,L) > 0. \end{aligned}$$

$\square$