Background: Let $X$ be a Fano variety over number field $K$, where its anticanonical bundle $K_X^{-1}$ is ample. Let $i: X \to \mathbb{P}^n$ be the anticanonical embedding, where $K_X^{-m} \cong i_*O(1)$. The standard (exponential) height $H$ on $\mathbb{P}^n$ is induced by standard metric on $O(1)$, and pull back to a height $i^*H$ on $X$.
Batyrev-Manin and Peyre conjectured a formula to count number of rational points of height up to $B$: $$N(X, K_X^{-1}, B) := \#\{x \in X(K): i^* H(x) \leq B\} \sim c(X, K_X^{-1}) B^{a(X, K_X^{-1}} (\log B)^{b(X, K_X^{-1})}$$ (Strictly speaking, one needs to focus on sufficiently small Zariski open subset of $X$, after removing accumulating subvarieties)
Batyrev-Tschinkel extended the conjecture to a more general setup, where the datum of $(X, L)$ is considered: here $X$ is a projective variety over $K$, and $L$ some suitable ample line bundle. For my question here, I only care about strongly saturated, $L$-primitive varieties $X$ in their language; this effectively means I do not need to remove any accumulating subvarieties. In this case, one expects an asymptotic formula $$N(X, L, B) := \#\{x \in X(K): H_L(x) \leq B\} \sim c(X, L) B^{a(X, L)} (\log B)^{b(X, L)}$$ as $B \to \infty$, where
- $a(X, L)$ is (effectively) $$a(X, L) := \inf\{t \in \mathbb{Q}: tL + K_X \in \Lambda_{eff}(X)\}$$ (Definition 2.2.4)
- (Unimportant, including for completeness) $b(X, L)$ is (effectively)
$$b(X, L) := rank(Pic(X, L))$$
(Definition 2.3.11) Here if $L^{a(X, L)} \otimes K_X$ is thought of as an effective divisor with support on irreducible components $D_1, \cdots, D_l$, then $Pic(X, L) := Pic(X - \bigcup_{i=1}^l D_i)$.
- As a sidenote, there is a natural restriction map $\rho: Pic(X) \otimes \mathbb{R} \to Pic(X, L) \otimes \mathbb{R}$. We will use $\Lambda_{eff}(X, L)$ to denote the image of $\Lambda_{eff}(X)$ under $\rho$. (Definition 2.3.11)
- $c(X, L)$ is of the shape
$$c(X, L) = \frac{\gamma(X, L)}{a(X, L) (b(X, L) - 1)!} \delta(X, L) \tau(X, L)$$
(page 32, middle of step 4). Here the extra constants are:
- $\gamma(X, L)$ is some invariant for the triple $(Pic(X, L), Pic(X, L) \otimes \mathbb{R}, \Lambda_{eff}(X, L))$ (Definition 2.3.14)
- $\delta(X, L)$ is the cardinality of $H^1(Gal(\overline{K}/K), Pic(X, L))$ (Definition 3.4.3)
- $\tau(X, L)$ is some Tamagawa measure of $\overline{X(K)}$ in $X(\mathbb{A}_K)$. (Definition 3.3.10)
Question I am interested in compatibility of the conjecture for $(X, L)$ and $(X, L^{\otimes k})$ for some positive integer $k$, particularly on the leading constant term $c(X, L)$. By checking definition,
- I am pretty sure that the constants/invariants $b, \gamma, \delta, \tau, Pic$ are the same for $(X, L)$ and $(X, L^{\otimes k})$.
- I think that $a(X, L^{\otimes k}) = \frac{1}{k} a(X, L)$. This should then imply $$c(X, L^{\otimes k}) = \frac{1}{k} c(X, L)$$
- At the same time, the induced heights should satisfy $H_{X, L^{\otimes k}} = H_{X, L}^k$, which should then mean $$N(X, L^{\otimes k}, B) \sim N(X, L, B^{1/k})$$
But then there seems to be a compatibility issue in the leading constant term: the conjecture for $(X, L)$ would predict $$N(X, L, B^{1/k}) \sim c(X, L) B^{a(X, L)/k} (\log B)^{b(X, L)}$$ However, the conjecture for $(X, L^{\otimes k})$ would predict $$N(X, L, B^{1/k}) \sim N(X, L^{\otimes k}, B) \sim c(X, L^{\otimes k}) B^{a(X, L^{\otimes k})} (\log B)^{b(X, L^{\otimes k})} = \frac{1}{k} c(X, L) B^{a(X, L)/k} (\log B)^{b(X, L)}$$ which is off by a factor of $\frac{1}{k}$.
Where in my calculation was wrong? Thank you!
