Plane Cremona maps: Saturation and regularity of the base ideal

Abstract

One studies plane Cremona maps by focusing on the ideal theoretic and homological properties of its homogeneous base ideal ("indeterminacy locus"). The leitmotiv driving a good deal of the work is the relation between the base ideal and its saturation. As a preliminary one deals with the homological features of arbitrary codimension 2 homogeneous ideals in a polynomial ring in three variables over a field which are generated by three forms of the same degree. The results become sharp when the saturation is not generated in low degrees, a condition to be given a precise meaning. An implicit goal, illustrated in low degrees, is a homological classification of plane Cremona maps according to the respective homaloidal types. An additional piece of this work relates the base ideal of a rational map to a few additional homogeneous "companion" ideals, such as the integral closure, the µ-fat ideal and a seemingly novel ideal defined in terms of valuations.

Saturation of Jacobian ideals: Some applications to nearly free curves, line arrangements and rational cuspidal plane curves

“…The self duality of the graded S-module N(f ), see [22,30,34], and the Lefschetz type property mentioned above imply that n(f ) s = 0 exactly for s = σ(C), ..., T − σ(C). In other words, for a reduced curve C : f = 0, one has (1.5) indeg(N(f )) = σ(C) and end(N(f )) = T − σ(C).…”

“…in the notation from [22]. We set as in [1,7] ν(C) = max j {n(f ) j }, and introduce a new invariant for C, namely σ(C) = min{j : n(f ) j = 0}.…”

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

“…for j = 1, ..., m − 2 and some integers ǫ j ≥ 1. Using the same approach as for the proof of Proposition 2.1, namely that one must have m(f ) k = τ (C) for all large k, or even better using [22,Formula (13)], it follows that one has (2.5)…”

“…Use [10, Theorem 1.2], with the remark that the equality in (1) We also consider the invariant σ(C) = min{j : n(f ) j = 0} = indeg(N(f )), in the notation from [22]. The self duality of the graded S-module N(f ), see [22,27,31], implies that 3d), where the leftmost map is the same as in the resolution (2.1). In particular,…”

“…Assume now that m > 3 and consider the resolution of N(f ) obtained from the resolution (2.3) by applying [22,Proposition 1.3], namely the resolution 3(d−1)).…”