The Cone of Effective Divisors of Log Varieties after Batyrev

In [Bat92] Batyrev studied the cone of pseudo-effective divisors on Q-factorial terminal threefolds and its dual cone, the cone of nef curves. Given a uniruled Q-factorial terminal threefold X , and an ample divisor H on X , he showed that the effective threshold of H (see Definition 1.5 below) is a rational number. Using similar arguments, Fujita generalized this result to log terminal pairs (X,∆), with dimX = 3 (see [Fuj96]). In [Bat92], using the rationality of the effective threshold and the minimal model program, Batyrev obtained a structure theorem for the cone of nef curves on Q-factorial terminal threefolds. We point out a problem in his proof of the structure theorem. Then we review the argument and use boundedness of terminal Fano threefolds to finish proof. Because of the use of this boundedness result, as it stands, this proof does not generalize to the log terminal case, as it has been claimed in previous papers. In section 1 we recall the main definitions and results of the minimal model program. In section 2 we explain Batyrev’s proof of the rationality of the effective threshold. We choose to work in great generality. We consider log terminal pairs (X,∆), with dimX = n. Then we assume the log minimal model program in dimension n, and obtain the rationality of the log effective threshold. In section 3 we explain the problem in Batyrev’s proof of the structure theorem for the cone of nef curves. In section 4 we provide a complete proof of this theorem (generalized to the case of terminal pairs). We work over some fixed algebraically closed field of characteristic zero.