  In the projective plane PG(2,q) over a finite field of order q, a Tallini
curve is a plane irreducible (algebraic) curve of (minimum) degree q+2
containing all points of PG(2,q). Such curves were investigated by G. Tallini
in 1961, and by Homma and Kim in 2013. Our results concern the automorphism
groups, the Weierstrass semigroups, the Hasse-Witt invariants, and quotient
curves of the Tallini curves.
