Is $M_g$ NEVER proper? And why does $T_g$ contain products? Work over an algebraically closed field ($\mathbb C$, if you prefer) and fix $g\geq 2$. By $M_g$ I mean, of course, the moduli space of smooth projective curves of genus $g$. I know it is in general not proper.
Question 1. Is it non proper for every $g$? And if not, what is an example of a proper $M_g$? 
The motivation for the above question is an argument I read on some lecture notes. This is how I understood it: let us consider the (injective) Torelli morphism $M_g\to A_g$ (defined by $[C]\mapsto [J(C)]$), where $A_g$ is the moduli space of PPAVs. Then, if $M_g$ were proper, its image would coincide with the Torelli locus $T_g\subset A_g$. But $T_g$ contains products of PPAVs, and no such product can be the Jacobian of a curve. Contradiction.
Now I'm lost.
Question 2,3. How to see that $T_g$ contains products of PPAVs? And why a product is not the Jacobian of any curve?
Also, in passing, where can I find a clean definition of the theta divisor on a PPAV? I am confused about this point.
Thanks in advance.
 A: Question 1:  $M_g$ ($g\ge 2$)  is never proper. A quick explanation is that $M_g$ is irreducible and has a compactification whose boundary is given by the singular stable curves of genus $g$. The boundary is non-empty because for any $g\ge 2$, there exists a singular stable curve of genus $g$ (attach a smooth projective curve of genus $1$ transversally to a smooth projective of genus $g-1$), so $M_g$ is different from its compactification, hence is not proper. 
A more elementary way to see the non-properness is to use the valuative criterion. Consider the  hyperelliptic curve of genus $g$ defined by 
$$ y^2=(x^2-t)(x^{2g}-1)$$
over $\mathbb C((t))$. Then over any finite extension of $\mathbb C[[t]]$ (necessarily of the form $\mathbb C[[t^{1/n}]$), the morphism 
$$\mathrm{Spec}( \mathbb C((t^{1/n})))\to M_g $$
can't extend to $\mathrm{Spec}(\mathbb C[[t^{1/n}]])$, because the curve
$$ y^2=(x^2-t_n)(x^{2g}-1), \quad t_n=t^{1/n}$$
has bad reduction. We have to prove this for all finite extension of $\mathbb C[[t]]$ because $M_g$ is a coarse moduli space. 
Question 2: If $C_1, ..., C_n$ are projective smooth curves of positive genus, form a chain with $C_i$ intersecting transversally $C_{i+1}$ at exactly one point. The stable curve $C$ obtained this way can be deformed to a smooth projective $X$. The Jacobian $C$ (equal to the product of the Jacobians of the $C_i$'s) deformes to the Jacobian of $X$. This shows that $\prod_i \mathrm{Jac}(C_i)$ is a limit of Jacobians, hence belongs to $T_g$. 
For the rest of the questions, I don't have a satisfactory answer to offer. 
