The miniversal deformation of certain complete intersection monomial curves

Abstract The aim of this paper is to provide an explicit basis of the miniversal deformation of a monomial curve defined by a free semigroup—these curves make up a notable family $$\mathscr {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> of complete intersection monomial curves. First, we dispense a general decomposition result of a basis B of the miniversal deformation of any complete intersection monomial curve. As a consequence, we explicitly calculate B in the particular case of a monomial curve defined from a free semigroup. This direct computation yields some estimates for the dimension of the moduli space of the family $$\mathscr {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> .