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5

In genus two the situation is very simple. All toroidal compactifications of $A_2$ are isomorphic and the DM compactification $\overline M_2$ is a toroidal compactification. I don't know a good reference unfortunately.

7

This map is usually called the Torelli map, not the Abel-Jacobi map. In any case, Mumford observed that a certain toroidal compactification of $\mathscr{A}_g$ admits an extension of the Torelli map; the original reference is this paper of Namikawa, I think. That paper doesn't give a very good moduli description of the map; luckily Alexeev does in this ...

2

Perhaps you could try using Groebner bases. The two examples that I computed using Macaulay2 (displayed below) suggest that there is a Groebner basis for $(f_n, f_{n+1})$ consisting of polynomials with leading terms of degree $n$. (These are $f_n, yf_{n-1}, y^2f_{n-2}, \cdots, y^n$, up to signs.) The examples also suggest that when we start reducing ...

2

The only natural thing I can think of is the following: Let $R_{\mathcal{P}}$ be the ring of integers of $K_{\mathcal{P}}$ (the completion at the prime). Let $S$ be the spectrum of $R_{\mathcal{P}}$. Since $A$ has good reduction, there is an abelian scheme (the Néron model) $\mathcal{A}$ over $S$, with $A$ as generic fibre, and the reduction as special ...

0

There is no such natural map. Note that both the tangent spaces you are talking about are vector spaces over different fields.

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