Rational and elliptic surface singularity of fiber type

A connected curve contained in a singular fiber of a $P^1$-fibration is known to be contractible to a rational singular point (see Lemma 1.1). We are interested in characterizing some specific properties of rational singularities of this type. We also show that a proper connected curve of a relatively minimal singular fiber of a genus $g>0$ fibration contracts to a singularity of arithmetic genus less than g. In particular, for $g=2$ we get either rational or elliptic singularities.