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Added definition of the Faltings height to answer Junkie's comment

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edited Oct 14 2011 at 7:20

Let $Y$ be a smooth projective connected curve of genus $g>0$ over $\overline{\mathbf{Q}}$. Let $h_{\textrm{Fal}}(Y)$ be the Faltings height of $Y$.

Question 1. Can one classify or describe the curves $Y$ such that $h_{\textrm{Fal}}(Y) \geq 1$?

Question 2. For any $g>0$, does there exist a curve $Y$ of genus $g$ such that $h_{\textrm{Fal}}(Y) <1$?

Essentially, I would like to know which curves one is excluding by looking at curves $Y$ such that $h_{\textrm{Fal}}(Y) \geq 1$.

A result of Bost says that the stable Faltings height of an abelian variety $A$ over $\overline{\mathbf{Q}}$ of dimension $g$ is bounded from below by $-\frac{1}{2}\log(2\pi)g$.

By the Northcott property of the Faltings height, the set of curves of genus $g$ with $h_{\textrm{Fal}}(Y) <1$ is finite. This means that I'm looking at the finite set of curves of genus $g$ with Faltings height not in the interval $$[-\frac{1}{2}\log(2\pi)g,1)\subset[-2/5g, 1).$$

Added: To answer Junkie's question, I'm aware of only one definition of the Faltings height of a curve over $\overline{\mathbf{Q}}$ . There are several equivalent definitions, though.

Let $X$ be a smooth projective curve of genus $g>0$ over $\overline{\mathbf{Q}}$. Let $K$ be a number field such that $X$ has a semi-stable regular model $p:\mathcal{X}\to \mathrm{Spec} O_K$ over the ring of integers $O_K$ of $K$. Then, the Faltings height $h_{\mathrm{Fal}}(X)$ of $X$ is the arithmetic degree $$h_{\mathrm{Fal}}(X):=\frac{\widehat{\mathrm{deg}} Rp_\ast \mathcal{O}_{\mathcal{X}}}{[K:\mathbf{Q}]},$$ where we endow the determinant of cohomology with the Arakelov-Faltings metric. This is well-defined, i.e., independent of the field $K$. By Serre duality, it coincides with $$h_{\mathrm{Fal}}(X)=\frac{\widehat{\mathrm{deg}} p_\ast \mathcal{\omega}_{\mathcal{X}/O_K}}{[K:\mathbf{Q}]}.$$ It also coincides with the Faltings height of the Jacobian. All of this is explained in Section 4.4 of

http://www.math.univ-toulouse.fr/~couveig/book.htm

For a curve over a number field, there is also the important relative Faltings height. This invariant depends on the number field $K$, though.

Changed the formulation of the question

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edited Oct 13 2011 at 22:10

Ariyan Javanpeykar
4,032●5●28

Let $Y$ be a smooth projective geometrically connected curve of genus $g$ g>0$ over a number field $K$. \overline{\mathbf{Q}}$. Let $h_{\textrm{Fal}}(Y)$ be the stable Faltings height of $Y$, i.e., the stable Faltings height of the Jacobian of $Y$. Suppose that $h_{\textrm{Fal}}(Y) \geq 1$. Then $g>0$.

Question 1. Can one classify or describe the curves $Y$ such that $h_{\textrm{Fal}}(Y) \geq 1$?

Question 2. For any $g>0$, does there exist a curve $Y$ of genus $g$ such that $h_{\textrm{Fal}}(Y) <1$? This question is already interesting to me when $g=1$.

Essentially, I would like to know which curves one is excluding by looking at curves $Y/K$ Y$ such that $h_{\textrm{Fal}}(Y) \geq 1$.

A result of Bost says that the stable Faltings height of an abelian variety $A/K$ A$ over $\overline{\mathbf{Q}}$ of dimension $g$ is bounded from below by $-Bg$ for some absolute constant $B$. -\frac{1}{2}\log(2\pi)g$.

Noted that B can be taken equal to \log(\pi \sqrt{2})

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edited Aug 13 2011 at 16:11

Ariyan Javanpeykar
4,032●5●28

Let $Y$ be a smooth projective geometrically connected curve of genus $g$ over a number field $K$. Let $h_{\textrm{Fal}}(Y)$ be the stable Faltings height of $Y$, i.e., the stable Faltings height of the Jacobian of $Y$. Suppose that $h_{\textrm{Fal}}(Y) \geq 1$. Then $g>0$.

Question 1. Can one classify or describe the curves $Y$ such that $h_{\textrm{Fal}}(Y) \geq 1$?

Question 2. For any $g>0$, does there exist a curve $Y$ of genus $g$ such that $h_{\textrm{Fal}}(Y) <1$? This question is already interesting to me when $g=1$.

Essentially, I would like to know which curves one is excluding by looking at curves $Y/K$ such that $h_{\textrm{Fal}}(Y) \geq 1$.

A result of Bost says that the stable Faltings height of an abelian variety $A/K$ of dimension $g$ is bounded from below by $-Bg$ for some absolute constant $B$. By the Northcott property of the Faltings height, the set of curves of genus $g$ with $h_{\textrm{Fal}}(Y) <1$ is finite. This means that I'm looking at the finite set of curves of genus $g$ with Faltings height not in the interval $[-Bg,1)$. (One can take $B=\log (\pi \sqrt{2})$ above.)