## The geometric Bogomolov conjecture for curves of small genus.(English)Zbl 1186.11035

Let $$K$$ be a function field of one variable over an algebraically closed field $$k$$. Let $$C$$ be a smooth proper geometrically connected curve of genus $$g \geq 2$$ over $$K$$. For a divisor $$D$$ of degree one on $$\bar{C} = C \times_K \bar{K}$$, one defines $a'(D) = \liminf_{x \in C(\bar{K})} \hat{h}([x] - D ),$ where $$\hat{h}$$ is the canonical Néron-Tate height on the Jacobian associated with the divisor $$\Theta + [-1]^* \Theta$$. The geometric Bogomolov conjecture claims that, if $$C$$ is not a constant curve, then one should have $$\inf_D a'(D) > 0$$, where $$D$$ runs all degree one divisor on $$\bar{C}$$. In this paper, the author verifies this conjecture when $$g \leq 4$$, by proving a more precise result explained below.
Let $$Y$$ be the smooth proper connected curve over $$k$$ with function field $$K$$. For simplicity, we assume $$C$$ admits a semistable reduction over $$K$$ and let $$f: X \to Y$$ be the minimal regular model of $$X$$. Take a point $$p \in X$$ which is a singular point of $$f^{-1}(f(p))$$. We call $$p$$ is of type zero (resp. of type $$i~ (1 \leq i \leq g/2))$$ if the partial normalization of $$f^{-1}(y)$$ at $$p$$ is connected (resp. has two components of arithmetic genus $$i$$ and $$g-i$$). Define $$\delta_i$$ to be the number of points on $$X$$ of type $$i$$. The author conjectures (and proves when $$g \leq 4$$) that the following should hold: $\inf_D a'(D) \geq \frac{1}{2(2g+1)} (\frac{g-1}{27g} \delta_0 + \sum_{1 \leq i \leq g/2} \frac{2i(g-i)}{g} \delta_i).$
The proof makes heavy use of a recent result of S.-W. Zhang [Invent. Math. 179, No. 1, 1–73 (2010; Zbl 1193.14031)]. The paper is very clearly written, and pleasant to read.

### MSC:

 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 11G50 Heights

Zbl 1193.14031
Full Text: