On logarithmic canonical divisors on threefolds (Q1076757)

Let V be a nonsingular projective 3-fold, defined over an algebraically closed field of characteristic zero and let \(D=D_ 1+...+D_ s\) be a divisor whose components are smooth and crossing normally on V. Let K be a canonical divisor of V. By extending an argument of \textit{P. M. H. Wilson} [Algebraic geometry, internat. Sympos. Centen. Birth F. Severi, Roma 1979, Sympos. Math. 24, 65-73 (1981; Zbl 0462.14010)] the author shows that if \(\kappa (V,K+D)\geq 0\), then \(K+D\) is ample if and only if it is numerically positive (i.e. \((K+D)\cdot C>0\) for all curves \(C\subset V)\). The statement remains true by replacing \(K+D\) with -K-D and this allows the author to rephrase the definition of logarithmic Fano 3-fold.