On Shimura curves in the Torelli locus of curves

Oort has conjectured that there do not exist Shimura curves lying generically in the Torelli locus of curves of genus $g\ge 8$. We show that there do not exist one-dimensional Shimura families of semi-stable curves of genus $g\ge 5$ of Mumford type. We also show that there do not exist Shimura curves lying generically in the Torelli locus of hyperelliptic curves of genus $g\ge 8$. The first result proves a slightly weaker form of the conjecture for the case of Shimura curves of Mumford type. The second result proves the conjecture for the Torelli locus of hyperelliptic curves. We also present examples of Shimura curves contained generically in the Torelli locus of curves of genus $3$ and $4$.