Plane curves with three syzygies, minimal Tjurina curves, and nearly cuspidal curves

Abstract

We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom-Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of curves.

On the minimal value of global Tjurina numbers for line arrangements

“…It is easy to see that the lower bound is strict, i.e. there are curves C of type (d, r) such that τ (C) = τ (d, r) min for any degree d and 1 ≤ r ≤ d − 1, see [17,Example 4.5] in general and [19,Lemma 21] for (d, r) = (d, d − 2). The curves C for which the maximal values for τ (C) are attained have special properties.…”

Quasi-complete intersections in $${\mathbb {P}}^2$$ and syzygies

“…In this note, inspired by [6], instead of considering only d 1 , the minimal degree of a generator of H 0 * (E), we consider the full minimal resolution of this module. So we will assume that H 0 * (E) is minimally generated by m elements of degree…”

“…In this case one say that C is a free divisor ( [14], [1]) or, equivalently, that Σ is an almost complete intersection. The case m = 3 is handled in [6]. Here we deal with the general case m ≥ 3.…”

“…Following [6] we have: Definition 1. We will say that C is a m-syzygy curve if H 0 * (E) is minimally generated by m elements of degree…”

“…If m = 3 and d 1 + d 2 = d, following [6] one says that C is a plus one generated curve. We see that C is a plus one generated curve if and only if Z (of degree…”