Heights and Tamagawa measures on Fano varieties
Let \(V\) be a Fano variety over a number field \(K\) such that the anticanonical class \(-K_V\) is very ample. To every basis of the \(K\)-vector space \(H^0(V,{\mathcal O}_V(-K_V))\) one can naturally associate a height \({\mathbf h}\) on \(V\). For every non-empty open subset \(U\) of \(V\) one denotes by \(n_U(H)\) the cardinal of the set \(\{P\in U(K)\mid {\mathbf h}(P)\leq H\}\). Then Manin conjectured that the asymptotic behaviour of \(n_U(H)\) is of the form \(n_U(H)\sim C\cdot H\cdot\log^{t-1}(H)\), where \(t\) is the rang of the Picard group of \(V\), provided \(U\) is sufficiently small. The first aim of the paper is to refine this conjecture of Manin by giving a conjectural expression of the constant \(C\) occurring in this formula. The advantage of this is that the refined conjecture thus obtained becomes stable under products of varieties, and, in particular, the author is able to recover the previous constants found by \textit{S. H. Schanuel} [Bull. Soc. Math. Fr. 107, 433--449 (1979; Zbl 0428.12009)] (in the case of projective spaces), and by \textit{J. Franke}, \textit{Yu. I. Manin} and \textit{Y. Tschinkel} [Invent. Math. 95, 421--435 (1989; Zbl 0674.14012)] for generalized flag manifolds. In the last part of the paper the author proves that the refined Manin conjecture holds true in some significant special cases, namely for the del Pezzo surfaces obtained by blowing up one, two, or three points of the rational projective plane \(P^2_{\mathbb Q}\), as well as for the variety obtained by blowing up \(P^n_{\mathbb Q}\) along some subspaces of \(P^n_{\mathbb Q}\).
scientific article; zbMATH DE number 809309
scientific article