\ge n-1$ and the equality holds if and only if $D$ is a tree. We calculate: $D . (K_X+D) = \sum D_j^2 + \sum K_X . D_j + 2 \sum_{k < j} D_k . D_j$ $\ge \sum_j (2p_a(D_j)-2) + 2(n-1) \ge -2$. The lemma follows.

\[treelem\] Suppose that $X$ is a nonsingular projective surface with $\chi({{\mathcal{O}}}_X) = 1$ and $D$ ($\ne 0$) a connected reduced divisor such that $H^0(X, K_X + D) = 0$. Then the following statements are true.

$(1)$ $D$ is a connected rational tree.

$(2)$ Suppose further that $D$ supports a nef and big divisor (so $D$ is automatically connected). Then $\pi_1(X) = (1)$.

The Serre duality and Riemann-Roch theorem imply $0 = h^0(X, K_X +
D) = h^1(X, K_X + D) + \frac{1}{2}(K_X + D) . D + \chi({{\mathcal{O}}}_X) \ge
0 + (-1) + 1$ by Lemma \[rat\]. Thus $D. (K_X + D) = -2$ and hence $D$ is a connected rational tree by the same lemma. So $\pi_1(D) = (1)$. Suppose that $D$ supports on a nef and big effective divisor. Then the surjective map $\pi_1(D) \rightarrow
\pi_1(X)$ in Nori \[No, Cor. 2.3\] infers $\pi_1(X) = (1)$.

The next result is a very important restriction 