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 


<https://arxiv.org/abs/math/0605244>


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



Which curves have stable Faltings height greater or equal to 1
Dear Ariyan, the elliptic curve with equation $$y^2=x^3+6$$ has Faltings height 
$$-(3/2)\log(\Gamma(1/3)/\Gamma(2/3))+(1/4)\log(3)=-0.748752...;$$ the curve 
of genus $2$ with equation $$y^2+y=x^5$$ has Faltings height 
$$
h\_{\rm Fal}(C\_{\bar{\bf Q}})=2\log(2\pi)-
{1\over 2}\log\big(\Gamma(1/5)^5\Gamma(2/5)^3\Gamma(3/5)\Gamma(4/5)^{-1}\big)
$$
$$
\approx
-1.452509239645644650317707042;
$$
For the first example, see Deligne, "Preuve des conjectures de Tate et Shafarevich", Séminaire Bourbaki. For the second one, see Bost, Mestre, Moret-Bailly, "Sur le calcul explicite des 'classes de Chern' des surfaces arithmétiques de genre $2$", Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). Astérisque No. 183 (1990), 69–105. 


Another explicit formula that should allow you to produce elliptic curves of arbitrarily large 
Faltings height is the inequality
$$
|h(j\_E)-12h\_{\rm Fal}(E)|\leqslant 6\log(1+h(j\_E))+47.15
$$
See paragraph 5. of the article "Serre's uniformity..." by Bilu and Parent for references.


Something else you can do is make numerical experiments with formula in Conj. 3 of the article of Colmez, "Hauteur de Faltings..." (Compositio), which is true (without $\log(2)$ factor !, see A. Obus, arXiv:1107.0684) if the CM field is abelian over $\bf Q$. In that case, the Artin $L$-functions become Dirichlet $L$-functions and can be computed explicitly in terms of values of the Gamma function using the Hurwitz formula.


This is not a complete answer but I hope that it helps.


