On the multiplicity of tangent cones of monomial curves (Q1741634)

One of the important problems in commutative algebra and algebraic geometry is to determine formulas and inequalities on the multiplicity of a standard graded algebra in terms of other numerical invariants, such as the codimension or degree of the defining equations. In the present paper the author concentrates on the standard graded algebra arising from a numerical semigroup, namely its tangent cone \(\mathcal{T}\). The author provides a sharp upper bound on the multiplicity of the tangent cone in terms of the codimension and the maximum degree of the equations of \(\mathcal{T}\). As a special case, he gives a complete affirmative answer to a question of \textit{J. Herzog} and \textit{D. I. Stamate} [Kyoto J. Math. 57, No. 3, 585--612 (2017; Zbl 1390.13014)].