**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](https://math.nyu.edu/%7Etschinke/papers/yuri/97dec/dec97.pdf) 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!



Leading constant in Batyrev-Tschinkel's refinement of Manin conjecture
Your calculation is off in a couple places but neither removes the discrepancy.


First, the exponent of $(\log B)$ is $b(X,L)-1$.


Second, substituting $B^{1/k}$ for $B$ in the term $(\log B)^{ b(X,L)-1}$ produces $$(\log B^{1/k} )^{ b(X,L)-1}= (\log B)^{ b(X,L)-1} / k ^{ b(X,L)-1}$$ instead of $(\log B)^{ b(X,L)-1} $.


Plugging this into your formulas, we get a discrepancy of $\frac{1}{k^{b(X,L)}}$ rather than $\frac{1}{k}$. This discrepancy would be exactly cancelled if $[\mathcal L]$ were used instead of $[K\_X]$ in the definition of $\gamma\_{\mathcal L}(V)$. So I wonder if this is what's meant.


