Points of bounded height on smooth hypersurfaces of toric varieties (Q2787075)

This paper uses the circle method to investigate Manin's conjecture for subvarieties of codimension one within certain toric varieties. It is inspired by the work of \textit{D. Schindler} [J. Reine Angew. Math. 714, 209--250 (2016; Zbl 1343.11061)], who considered hypersurfaces in bi-projective space. Schindler's work itself built on ideas of \textit{B. J. Birch} [Proc. R. Soc. Lond., Ser. A 265, 245--263 (1962; Zbl 0103.03102)], which used the circle method.NEWLINENEWLINEThe toric variety considered here is the quotient of NEWLINE\[NEWLINE(\mathbb{A}^{r+1}\setminus\{0\})\times((\mathbb{A}^{m-r}\times \mathbb{A}^{n-m+1})\setminus\{0\})\subset\mathbb{A}^{n+2}NEWLINE\]NEWLINE by the action of the torus \(\mathbb{C}^*\times \mathbb{C}^*\), given by NEWLINE\[NEWLINE(\lambda,\nu)\cdot(\mathbf{x},\mathbf{y},\mathbf{z})= (\lambda\mathbf{x},\lambda\mu\mathbf{y},\mu\mathbf{z}).NEWLINE\]NEWLINE One then has a hypersurface \(Y\) in this toric variety, given by \(F(\mathbf{x},\mathbf{y},\mathbf{z})=0\), where \(F\) is homogeneous of degree \(d_1\), say in \((\mathbf{x},\mathbf{y})\) and homogeneous of degree \(d_2\) in \((\mathbf{y},\mathbf{z})\).NEWLINENEWLINEFor an appropriate height function on \(Y\) and a suitable open subset \(U\), both specified in the paper, it is then shown that Manin's conjecture holds in the form NEWLINE\[NEWLINEN_{U,H}(B)=C_H(Y)B\log B+O(B)NEWLINE\]NEWLINE for large enough \(n\). Here \(C_H(Y)\) is the constant predicted by \textit{E. Peyre} [Duke Math. J. 79, No. 1, 101--218 (1995; Zbl 0901.14025)]. The condition required for \(n\) is that NEWLINE\[NEWLINEn+2-\max\{\dim V_1^*,\text{dim}\, V_2^*\}>13d_2(d_1+d_2)2^{d_1+d_2},NEWLINE\]NEWLINE provided that \(d_1\geq 2\) and \(d_1\geq 1\). Here \(V_1^*\) and \(V_2^*\) are certain singular loci, in the spirit of Birch's work.NEWLINENEWLINEThe difficulty with the proof is that the condition \(H(\mathbf{x},\mathbf{y},\mathbf{z})\leq B\) does not correspond to a rectangular box in \(\mathbb{R}^{n+2}\). As a result one has to count points in boxes defined by two parameters \(P_1\) and \(P_2\), and combine the various results, using the ideas of \textit{V. Blomer} and \textit{J. Brüdern} [``Counting in hyperbolic spikes : the diophantine analysis of multihomogeneous diagonal equations'', Preprint, \url{arXiv:1402.1122}]. There are different treatments for the cases in which \(P_1\) and \(P_2\) are ``small'', and that in which they have comparable size. The overall framework follows that of Schindler, but there are many additional technicalities.