On freeness of divisors on \mathbb P²

2012/03/09 by Tohaneanu, Stefan O.
#13C14 (Secondary) #13D02 (Primary) 52C35 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1203.2046

Abstract

Let I⊂ \mathbb C[x,y,z] be an ideal of height 2 and minimally generated by three homogeneous polynomials of the same degree. If I is a locally complete intersection we give a criterion for \mathbb C[x,y,z]/I to be arithmetically Cohen-Macaulay. Since the setup above is most commonly used when I=J_F is the Jacobian ideal of the defining polynomial of a "quasihomogeneous" reduced curve Y=V(F) in \mathbb P², our main result becomes a criterion for freeness of such divisors. As an application we give an upper bound for the degree of the reduced Jacobian scheme when Y is a free rank 3 central essential arrangement, as well as we investigate the connections between the first syzygies on J_F, and the generators of √(J_F).

Related

• Jacobian schemes arising from hypersurface arrangements in \mathbb Pⁿ
• Free divisors, blowup algebras of Jacobian ideals, and maximal analytic spread
• A result on Macaulay's curve
• Schemes supported on the singular locus of a hyperplane arrangement in \mathbb Pⁿ
• A family of irreducible free divisors in P²