Weierstrass semigroups on Castelnuovo curves

We define a class of numerical semigroups S, which we call Castelnuovo semigroups, and study the subvariety \(M^S_{g,1}\) of \(M_{g,1}\) consisting of marked smooth curves with Weierstrass semigroup S. We determine the number of irreducible components of these loci and determine their dimensions. Curves with these Weierstrass semigroups are always Castelnuovo curves, which provides the basic tool for our argument. This analysis provides examples of numerical semigroups for which \(M^S_{g,1}\) is reducible and non-equidimensional.