Counting points on a toric variety (Q5931324)

The author proves the following result: if \(V\) is a smooth split toric variety over \({\mathbb Q}\) and \(-K_V\) is generated by its global sections then NEWLINE\[NEWLINE N_T(B)=BQ(\log B)+O(B^{1-\theta}) NEWLINE\]NEWLINE for some \(\theta>0\). Here \(T\) is the maximal torus in~\(V\), \(Q\) is a polynomial of degree \(\rho(V)-1\) and \(N_T(B)\) is the number of points of \(T({\mathbb Q})\) of height \(\leq B\). Moreover the leading coefficient of \(Q\) agrees with that predicted by \textit{E. Peyre} in [Duke Math. J. 79, 101-218 (1995; Zbl 0901.14025)]. This theorem is a sharpening of results of \textit{A. Batyrev} and \textit{Yu. Tschinkel} [J. Algebr. Geom. 7, 15-53 (1998; Zbl 0946.14009)] and of \textit{P. Salberger} [Astérisque 251, 91-258 (1998; Zbl 0959.14007)] NEWLINENEWLINENEWLINEThe proof uses a method due to Salberger [loc. cit.] to reduce the problem to an estimation of the asymptotic behaviour of a sum NEWLINE\[NEWLINE \sum_{1\leq k_i \leq B} f(k_1,\ldots,k_m) NEWLINE\]NEWLINE where \(f\) is a certain multiplicative function defined by the fan of \(V\). This estimation is immediately achieved by applying a result of the author [Compos. Math. 128, 261-298 (2001; Zbl 1001.11036)]. The paper concludes with a detailed study of one example, a Del Pezzo surface of degree 6.