ON THE IRREDUCIBILITY OF SECANT CONES, AND
... W is normal, then so is a general fibre of h: this can be seen, e.g. using
Serre’s criterion, or alternatively, use [G], Thm 12.2.4, which says, in the
scheme-theoretic context, that the set N (h) of points t ∈ T such that h−1 (t)
is normal is open; when W is normal, N (h) contains the generic point ...