I would like to understand the application of the Hardy-Littlewood Circle Method to Birch's celebrated theorem on forms in many variables, namely the statement that the point counting function for a smooth projective hypersurface of degree $d$ in $\mathbb{P}^N$ defined over $\mathbb{Q}$ satisfies the usual asymptotic if $N\gg_d 1$ (and in particular that such surfaces satisfy the Hasse principle). I am following an exposition of the quartic case in Timothy Browning's Quantitative Arithmetic of Projective Varieties, Chapter 8. Unless I am highly mistaken, the proof presented there requires at least some modifications, and I would like to confirm whether I am right when I say this.
My question concerns the minor arcs estimate. This depends on Lemma 8.7, which gives an estimate $$S\left(\frac{a}{q}+z\right)\ll_\varepsilon B^{n+\varepsilon}q^{-\frac{n}{24}}\min\{1,|z|B^4\}^{-\frac{n}{24}}$$ for the relevant exponential sum. This yields (at the end of section 8.2.1) the estimate $$\int_{\mathfrak{m}(\Delta)}|S(\alpha)|\mathrm{d}\alpha\ll_\varepsilon\sum_{q\leq B^2}q^{1-\frac{n}{24}}\int_{-\frac{1}{qB^2}}^{\frac{1}{qB^2}}\min\{1,|z|B^4\}^{-\frac{n}{24}}\mathrm{d}z(*)$$ from where the author concludes (using that we are actually only summing over values of $q$ and $z$ that are admissible for the minor arcs; in particular this includes all $q>B^\Delta$) that the contribution of the minor arcs is $\ll_\varepsilon B^{n-4-\Delta(\frac{n}{24}-2)+\varepsilon}$, which beats the desired $B^{n-4}$ if $n>48$.
But I don't see how this follows from $(*)$, and in fact, unless I'm very much mistaken, it can't follow from $(*)$: if $n>48$, the integral that appears on the RHS of $(*)$ explodes near $0$, since the $\min$ is actually $|z|B^4$, for small $z$! (And we actually need to care about small $|z|$ for some minor arcs, namely those corresponding to $q>B^\Delta$!)
Does my concern make sense? If so, how can we fix this proof? I am particularly interested in understanding exactly how the lower bound $48$ appears in the proof, so understanding these technical details is important. Thanks in advance!
