On the minimal free resolution of the Rees algebra of some monomial ideals

Abstract We give a complete description of the defining ideal $$\mathcal {R}(I)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of the Rees algebra of any monomial ideal I minimally generated by three monomials in two variables. We give a Gröbner basis and a complete explicit description of the minimal free resolution of $$\mathcal {R}(I)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> based on arithmetical properties of the generators of the ideal I . We also present algorithms for the computation of $$\mathcal {R}(I)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and its minimal free resolution. These results extend and generalize previous work by Cox and by Cortadellas and D’Andrea on parametrizations of monomial plane curves.