On the Cohen-Macaulay Property for Quadratic Tangent Cones

Let H be an n-generated numerical semigroup such that its tangent cone grmK[H] is defined by quadratic relations. We show that if n < 5 then grmK[H] is CohenMacaulay, and for n = 5 we explicitly describe the semigroups H such that grmK[H] is not Cohen-Macaulay. As an application we show that if the field K is algebraically closed and of characteristic different from two, and n 6 5 then grmK[H] is Koszul if and only if (possibly after a change of coordinates) its defining ideal has a quadratic Gröbner basis.