Proof. Put \(K=\mathbb C(Y)\) and \(\eta=\operatorname{Spec}K\). Generic smoothness and Stein factorization give a smooth geometrically integral generic fiber \(X_\eta\) (Hartshorne 1977, III, Sections 10–11). The cotangent sequence along this fiber gives \[ \omega_X|_{X_\eta} \simeq\omega_{X_\eta/K}\otimes_K\ell, \qquad \ell=\omega_Y|_\eta. \tag{3}\] Here \(\ell\) is a one-dimensional \(K\)-vector space. The factor from the base therefore cancels in ratios of pluricanonical sections.

Fix \(m>0\) such that \(H^0(X,mK_X)\ne0\), and choose a basis \(s_0,\ldots,s_N\) with \(s_0\ne0\). A nonzero section is nonzero at the generic point of \(X\), so it stays nonzero on \(X_\eta\) and after extension to the geometric generic fiber. By (3), the restricted system on \(F\) is a nonzero subsystem of \(|mK_F|\). Its image has dimension zero, since \(\kappa(F)=0\).

Let \(\phi_m\) be the rational map of the complete system \(|mK_X|\), and let \(T\) be the closure of the image of \((f,\phi_m)\colon X\dashrightarrow Y\times\mathbb P^N\). Its function field is \[L=K(s_1/s_0,\ldots,s_N/s_0)\subseteq\mathbb C(X).\] The dimension of its generic image over \(Y\) is \(\operatorname{trdeg}_K L\). This dimension is unchanged by algebraic extension of \(K\). Geometric integrality of \(X_\eta\) identifies that extended image with the image of the same section ratios on \(F\). Thus \(\operatorname{trdeg}_K L=0\), and \(\dim T=\dim Y\). Projection to \(\mathbb P^N\) gives \(\dim\phi_m(X)\leq\dim Y\). Taking the maximum over all nonempty pluricanonical systems proves the lemma. If every system is empty, the inequality is immediate. ◻

Proof. The inequality \(k<y\leq d\) makes all exponents nonnegative; a product with no factors has its usual meaning. We prove first that the test is nonnegative, then that it has the two required vanishing properties, and finally that it detects the ample divisor from \(Y\).

Nonnegativity. The divisors \(q^*D\) and \(q^*A\) are globally generated. If \(B\) is effective, choose their members successively so that no member contains an irreducible component of the preceding intersection on \(B\). Global generation permits all the finitely many required avoidances. The resulting Cartier intersections form an effective zero-cycle, possibly empty. Hence \(B\cdot C\geq0\) (Fulton 1998, chap. 2). Extend by linearity to effective real divisors and by continuity on \(N^1(W)_\mathbb R\) to its closed effective cone. This proves (1).

Vanishing on exceptional and semiample classes. For a \(q\)-exceptional divisor the Cartier projection formula gives \[ E\cdot(q^*D)^k(q^*A)^{d-k-1} =q_*E\cdot D^kA^{d-k-1}=0; \tag{4}\] see (Fulton 1998, Proposition 2.3(c)). Indeed, every component of \(E\) maps to codimension at least two, so \(q_*E=0\) as a divisor cycle. This reasoning is valid on the possibly singular variety \(V\). Also \(D=g^*\mathcal O_Z(1)\), and \(k+1\) general hyperplanes miss the \(k\)-dimensional image \(Z\). Consequently \((q^*D)^{k+1}=0\). For \(k=0\), the morphism \(g\) is constant and \(\mathcal O_V(D)\) is trivial, which gives the same conclusion. These observations prove (2) and the first assertion of (3).

Strict positivity. Choose \(b>0\) such that \(bH\) is very ample. Use this divisor to embed \(Y\), and use \(A\) to embed \(V\). There is a point \(w\in W\) at which \(q\) is an isomorphism onto a smooth open subset of \(V\) and \[\operatorname{rank}d(gq)_w=k, \qquad\operatorname{rank}dh_w=y, \qquad\operatorname{rank}dq_w=d.\] Each condition holds on a dense open subset. For the rank statements, the generic differential rank in characteristic zero equals the dimension of the image, by separability of the function-field extension. We choose \(w\) in the intersection of these open subsets.

The differentials of pulled-back hyperplanes through \(gq(w)\) span a \(k\)-dimensional subspace \(U\subseteq T_w^*W\). Choose \(k\) of them whose differentials form a basis of \(U\). The hyperplanes through \(h(w)\) supply a \(y\)-dimensional subspace. Since \(y>k\), one such hyperplane pulls back to a divisor with differential outside \(U\). Finally, hyperplanes from the embedding by \(A\) supply all cotangent directions at \(w\), because \(q\) is a local isomorphism there. Choose \(d-k-1\) of them completing the chosen differentials to a basis of \(T_w^*W\). These \(d\) divisors meet transversely at \(w\).

To use this local intersection in the global intersection number, perturb the hyperplanes slightly in their complex parameter spaces. The transverse point persists for every sufficiently small perturbation, by the implicit function theorem. Tuples whose pullbacks meet properly at every successive step form a nonempty Zariski open subset of the product of the hyperplane parameter spaces. Nonemptiness follows by successively avoiding the finitely many components already present, using global generation; openness follows from upper semicontinuity of fiber dimension for the projective universal intersections. This parameter space is irreducible, so the open subset is dense also in the complex analytic topology. It therefore meets the perturbation neighborhood.

The resulting proper global intersection is an effective zero-cycle that contains a transverse point of multiplicity one. It follows that \[h^*(bH)\cdot(q^*D)^k(q^*A)^{d-k-1}>0.\] Division by \(b\) proves (3). Only one cotangent direction from \(h\) outside \(U\) was needed; the two morphisms \(gq\) and \(h\) need not factor through each other. ◻

Proof of Theorem 4. If \(Y\) is a point, then \(F=X\) and \(\kappa(F)=0\) is the required conclusion. Hence assume \(d=\dim X\geq y=\dim Y>0\).

Let \(V\) be a good klt minimal model of \(X\). Resolve the closure of the graph of \(X\dashrightarrow V\) projectively (Hironaka 1964), obtaining a smooth projective common resolution \(X\xleftarrow{p}W\xrightarrow{q}V\). By the good-model comparison, compatible canonical divisors satisfy \[ p^*K_X=q^*K_V+E,\qquad E\geq0,\qquad q_*E=0. \tag{5}\] Pulling back to a further common resolution preserves the effectivity and target-exceptionality of this error.

Choose \(r>0\) such that \(D=rK_V\) is Cartier and globally generated, and let \(g\colon V\to Z\subseteq\mathbb P^N\) be its morphism onto its image. Put \(k=\dim Z\). Since \(K_X\) is Cartier, (5) shows that \(rE=p^*(rK_X)-q^*D\) is an integral Cartier divisor on \(W\). The two maps whose dimensions we will compare appear in Figure 1.

Pull back sections of \(D\) by \(q\) and multiply by the canonical section of the effective Cartier divisor \(rE\). We obtain an injection \[ H^0(V,D)\hookrightarrow H^0(W,rp^*K_X)=H^0(X,rK_X). \tag{6}\] The equality follows from \(p_*\mathcal O_W=\mathcal O_X\) and the sheaf projection formula for the proper birational map to the normal variety \(X\) (Hartshorne 1977, III, Corollary 11.4 and its proof; II, Exercise 5.1(d)). Multiplication by the section of \(rE\) leaves section ratios unchanged on the complement of its support. Thus the subsystem in (6) has image dimension \(k\), and \[ \kappa(X)\geq k. \tag{7}\]

Suppose, for a contradiction, that \(k<y\). Choose a very ample Cartier divisor \(A\) on \(V\) and apply Lemma 6 with \[h=fp,\qquad C=(q^*D)^k(q^*A)^{d-k-1}.\] The canonical comparison and the vanishing assertions of the lemma give \[ p^*K_X\cdot C =\frac1r q^*D\cdot C+E\cdot C=0. \tag{8}\] Pullback by \(p\) preserves pseudo-effectivity: on smooth varieties it takes effective real divisors to effective real divisors and induces a continuous linear map on numerical divisor spaces. Therefore the pullback of (2), paired with \(C\), yields \[0\leq(m_0p^*K_X-h^*H)\cdot C=-h^*H\cdot C<0,\] a contradiction. Hence \(k\geq y\). Equation (7) and Lemma 5 now give \(y\leq k\leq\kappa(X)\leq y\), proving the theorem. ◻

Proof of Corollary 2. Since \(D\) is effective, \(\kappa(D)\geq0\). Also \[K_X=\frac1{m_0}\bigl((m_0K_X-D)+D\bigr)\in\overline{\operatorname{Eff}}(X).\] If \(\kappa(D)=0\), canonical nonvanishing (9) proves the assertion. We may therefore assume \(\kappa(D)>0\).

Schnell’s reduction in (Schnell 2022, secs. 4–6) first resolves a system \(|nD|\) whose image has dimension \(\kappa(D)\), then takes its Stein factorization and resolves the base. It produces an algebraic fiber space \(f_0\colon X_0\to Y_0\) between smooth connected projective complex varieties, with \(X_0\) birational to \(X\), an ample Cartier divisor \(H_0\) on \(Y_0\), and a positive integer \(a_0\) such that \[ \dim Y_0=\kappa(D),\qquad a_0K_{X_0}-f_0^*H_0\in\overline{\operatorname{Eff}}(X_0). \tag{10}\] The effective exceptional terms from resolving \(X\) preserve pseudo-effectivity; after resolving the base, Schnell replaces the big and nef pullback of the original ample divisor by a suitable ample divisor. In particular, the output has the precise ample Cartier hypothesis of Theorem 1.

We recall the finite iteration in (Schnell 2022, sec. 7 and 9). Suppose that \(f_i\colon X_i\to Y_i\) is such a fiber space, with ample Cartier \(H_i\) and \(a_iK_{X_i}-f_i^*H_i\in\overline{\operatorname{Eff}}(X_i)\) for a positive integer \(a_i\). For a very general smooth fiber \(F_i\), the preceding restriction and nonvanishing argument gives \(\kappa(F_i)\geq0\). If \(\kappa(F_i)>0\), the construction in (Schnell 2022, Lemma 7.1) chooses positive integers \(r_i,b_i\) and an effective Cartier divisor \(L_i\) such that \[L_i\sim r_iK_{X_i}+f_i^*(b_iH_i),\qquad \kappa(L_i)=\kappa(F_i)+\dim Y_i.\] The resulting comparison is \[(a_ib_i+r_i)K_{X_i}-L_i \sim b_i(a_iK_{X_i}-f_i^*H_i).\] Its right side is pseudo-effective, so the same is true of its left side, since linear equivalence preserves numerical classes. Applying the divisor-to-fiber-space reduction of Sections 4–6 of Schnell’s paper to \(L_i\) gives another algebraic fiber space \(f_{i+1}\colon X_{i+1}\to Y_{i+1}\) of the same smooth projective complex type, with \(X_{i+1}\) birational to \(X_i\), an ample Cartier divisor \(H_{i+1}\), and a positive integer \(a_{i+1}\) such that \[\dim Y_{i+1}=\kappa(F_i)+\dim Y_i,\qquad a_{i+1}K_{X_{i+1}}-f_{i+1}^*H_{i+1}\in\overline{\operatorname{Eff}}(X_{i+1}).\]

As long as \(\kappa(F_i)>0\), the integer \(\dim Y_i\) strictly increases. It is bounded by \(\dim X\), so this process reaches an index \(j\) with \(\kappa(F_j)=0\). The geometric generic fiber has Kodaira dimension zero by the comparison above. Theorem 1 applies at this last stage. Birational invariance of Kodaira dimension and (10) give \[\kappa(X)=\kappa(X_j)=\dim Y_j \geq\dim Y_0=\kappa(D).\] ◻

Proof of Corollary 3. If \(Y\) is a point, then \(F=X\) and \(H\) is linearly equivalent to zero. The equality is tautological, and the numerical hypothesis gives pseudo-effective \(K_X\). A nonzero section supplied by (9), together with its powers, gives the asserted sections.

Assume \(\dim Y>0\). The restriction argument above gives \(\kappa(F)\geq0\). As in (Schnell 2022, Lemma 7.1), choose positive integers \(a,b\) and an effective Cartier divisor \(L\) with \[L\sim aK_X+f^*(bH),\qquad \kappa(L)=\kappa(F)+\dim Y.\] The linear equivalence \[(m_0b+a)K_X-L\sim b(m_0K_X-f^*H)\] shows that its left side is pseudo-effective. Corollary 2, applied to the effective Cartier divisor \(L\), gives \[\kappa(X)\geq\kappa(L)=\kappa(F)+\dim Y.\] The reverse inequality is the easy-addition inequality used in (Schnell 2022, Lemma 7.1). This proves the equality.

To obtain sections after subtracting the original \(f^*H\), we use the Fujita–Mori criterion recalled in (Popa and Schnell 2014, Lemma 4.6); this is the criterion behind (Schnell 2022, sec. 8). Since \(\kappa(F)\geq0\), the equality just proved implies that there are a big Cartier divisor \(B\) on \(Y\) and a positive integer \(u\) with a nonzero section \[\sigma\in H^0\bigl(X,\mathcal O_X(uK_X-f^*B)\bigr).\] Bigness of \(B\) gives a positive integer \(v\) and a nonzero section \(\tau\in H^0(Y,\mathcal O_Y(vB-H))\). Thus, with \(r=uv\), \[s=\sigma^v f^*\tau \in H^0\bigl(X,\mathcal O_X(rK_X-f^*H)\bigr)\] is nonzero. This step replaces the big divisor supplied by the criterion with the ample divisor in the original hypothesis.

Finally, ampleness of \(H\) supplies a nonzero section \(\tau_\ell\in H^0(Y,\mathcal O_Y((\ell-1)H))\) for every sufficiently large integer \(\ell\). The products \[s^\ell f^*\tau_\ell \in H^0\bigl(X,\mathcal O_X(\ell rK_X-f^*H)\bigr)\] are nonzero. Choosing \(\ell_0\) beyond this ampleness threshold proves the asserted conclusion for every integer \(\ell\geq\ell_0\). ◻