Cuspidal plane curves, syzygies and a bound on the MW-rank

Abstract

Let C = Z(f ) be a reduced plane curve of degree 6k, with only nodes and ordinary cusps as singularities. Let I be the ideal of the points where C has a cusp. Let ⊕S(−b i ) → ⊕S(−a i ) → S → S/I be a minimal resolution of I. We show that b i ≤ 5k. From this we obtain that the Mordell-Weil rank of the elliptic threefold W :Using this we find an upper bound for the Mordell-Weil rank of W , which is 1 18 (125 + √ 73 − 2302 − 106 √ 73)k + l.o.t. and we find an upper bound for the exponent of (t 2 − t + 1) in the Alexander polynomial of C, which is 1 36 (125 + √ 73 − 2302 − 106 √ 73)k + l.o.t.. This improves a recent bound of Cogolludo and Libgober almost by a factor 2.

On the syzygies and Alexander polynomials of nodal hypersurfaces

“…This study gives sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining a nodal hypersurface, extending the result proved in the nodal curve case in Theorem 4.1 in 11 to arbitrary dimension. In the curve case, see also 13 and 16.…”

Koszul Complexes and Pole Order Filtrations

“…In fact, a recent result by Kloosterman (see [24,Proposition 3.6]) implies that the first part of Corollary 1.6 holds for any curve C with the property that any singular point of C that is not a node is a unibranch singularity (see Remark 4.4 for more details on this).…”

“…Since N is a finite set of points, it follows that this series can be rewritten as Alternatively, one may end the proof using the formula for the defect or superabundance def S k (N ) as the difference between the Hilbert polynomial and the Hilbert function given in [24], just before the statement of Lemma 3.4.…”

On the Milnor Monodromy of the Irreducible Complex Reflection Arrangements

“…starting from V or f is a rather difficult problem, going back to O. Zariski and attracting an extensive literature, see for instance [2,9,11,19,21,23,24,29,37] for the case n = 2, and some of them dealing only with real line arrangements. In this paper, we take a new look at a method to determine the Alexander polynomial V (t) introduced in [8] and developed in [9,Chapter 6].…”