  Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete
valuation ring. Two fundamental numerical invariants are attached to X/S: the
valuation of the hyperelliptic discriminant of X/S, and the valuation of the
Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue
field of characteristic 0 as well as for X/S semistable these invariants are
known to satisfy certain inequalities. We prove an exact formula relating the
two invariants with intersection theoretic data determined by the distribution
of Weierstrass points over the special fiber, in the semistable case. We also
prove an exact formula for the stable Faltings height of an arbitrary curve
over a number field, involving local contributions associated to its
Weierstrass points.
