Graded derivation modules and algebraic free divisors

Free divisors, blowup algebras of Jacobian ideals, and maximal analytic spread

“…. Now we remark that, since T R/k (f ) decomposes into the direct sum of the module of syzygies of J f and the cyclic module Rε n (see [31,Lemma 2.2]), a free basis of T R/k (f ) if f is a free divisor consists of the derivations corresponding to the columns of a minimal syzygy matrix of f x 1 , . .…”

“…For example, ε n ∈ [T R/k (f )] 0 . Finally recall that there is an identification (see, e.g., [31,Lemma 2.1])…”

“…(iii) First, if α = 2, a simple inspection shows that the linear type property of J f fails in case β ≥ 3 (and holds if β = 2), by analyzing the saturation of the ideal S of 2 linear forms defining Sym R J f in R[s, t, u] by the ideal J f . The resulting ideal -which thus defines R(J f ) (see, e.g., [31,Lemma 2.11]) -turns out to strictly contain S . In addition, it is contained in (x, y, z)R[s, t, u], so that F (J f ) ∼ = k[s, t, u].…”

“…Over time, different approaches, viewpoints, and interests have emerged, including algebraic (and algebro-geometric) adaptations and even generalizations that have drawn the attention of an increasing number of researchers over the last fifteen years. The list of references is huge; see, e.g., Abe [1], Abe, Terao and Yoshinaga [2], Buchweitz and Conca [8], Buchweitz and Mond [9], Calderón-Moreno and Narváez-Macarro [12], Damon [15], Dimca [17,18], Dimca and Sticlaru [20], Miranda-Neto [31,32], Schenck [37], Schenck, Terao and Yoshinaga [38], Schenck and Tohǎneanu [39], Simis and Tohǎneanu [41], Tohǎneanu [47], and Yoshinaga [51]. In particular, nice references containing interesting open problems on the subject (including the celebrated Terao's Conjecture) are Dimca's book [17] and Schenck's survey [37].…”

“…For f ∈ R belonging to any of the four (or five) families, we use a result from Miranda-Neto [31] to easily determine the Castelnuovo-Mumford regularity of the graded module Der k (R/(f )) formed by the k-derivations of the ring R/(f ). Needless to say, the regularity is an important invariant which controls the complexity of a module (being related to bounds on the degrees of syzygies), whereas the derivation module is a classical object as it collects the tangent vector fields defined on the hypersurface V (f ) ⊂ P n−1 k .…”