Affine surfaces and projective curves

Let $$F \in \mathbb{Z}[x_0, x_1, x_2]$$ be a geometrically irreducible homogeneous polynomial of degree $$d \geq 3$$, so that the equation $$F = 0$$ defines a projective curve $$C_F$$ in $$\mathbb{P}^2$$ of degree $$d$$. We represent rational points in $$\mathbb{P}^2$$ the usual way, namely by a primitive integral triple $$(u_0, u_1, u_2)$$. For a rational (primitively integral) point $$\mathbf{u} \in \mathbb{P}^2$$, define the (naive) height to be $$H(\mathbf{u}) = \max\{|u_0|, |u_1|, |u_2|\}$$.

It is now known (due to Miguel Walsh) that the set of rational points of bounded height on $$C_F$$, given by

$$\displaystyle C_F(X) = \{\mathbf{u} \in C_F(\mathbb{Q}) : H(\mathbf{u}) \leq X\}$$

satisfies

$$\displaystyle |C_F(X)| \ll_d X^{2/d},$$

where the implied constant depends at most on $$d$$.

If we look at the analogous question for affine surfaces, defined by a polynomial $$G(x_0, x_1, x_2) \in \mathbb{Z}[x_0, x_1, x_2]$$ of degree $$d \geq 3$$ which is geometrically integral (but not necessarily homogeneous), then the affine surface $$S_G : \{G = 0\}$$ does not behave as nicely. Indeed, if we consider the set

$$\displaystyle S_G(X) = \{\mathbf{u} \in \mathbb{A}^3(\mathbb{Z}) \cap S_G : \mathbf{u} \text{ primitive}, H(\mathbf{u}) \leq X\}$$

then it would not be true that the Walsh bound holds. Indeed, looking at the example $$G = x_0^3 + x_1^3 + x_2^3 - 1$$ and the subvariety defined by $$x_0 = 1, x_1 = - x_2$$ shows that $$|S_G(X)| \gg X$$ even though $$\deg G = 3$$.

The problem here is that the cubic surface defined above contains a line, which contains many integral points. This problem is subtly removed in the projective curve case because the primitivity assumption means that this can't happen; if we allow imprimitive points and $$C_F$$ contains a single rational point, then $$C_F$$ would contain $$\gg X$$ integral points from just points proportional to a fixed point.

Suppose we attempt to remove this issue for affine surfaces as well; namely, we remove lower-dimensional subvarieties that could contain too many integral points. Indeed, for a given surface $$S_G$$ let $$U_G$$ be the open subset of $$S_G$$ obtained by removing all curves of degree at most $$d-2$$ from $$S_G$$ (it is known that $$S_G$$ contains at most finitely many of such curves). Define $$U_G(X)$$ analogously to $$S_G(X)$$. Can one expect

$$\displaystyle |U_G(X)| \ll_d X^{\theta_d}$$

for some $$\theta_d < 1$$, with implied constant depending only on $$d$$?

• en.wikipedia.org/wiki/Manin_conjecture – Felipe Voloch Dec 3 at 5:09
• Actually Manin's conjecture is only stated for smooth projective varieties, so does not strictly cover this case. But a similar philosophy should hold here. Quite possibly what you want is already known. The best results come from the determinant method and Salberger has the best results, but he is not very good at publishing them. I would just recommend emailing Salberger to ask him. – Daniel Loughran Dec 3 at 11:13