Addition-deletion results for the minimal degree of a Jacobian syzygy of a union of two curves

Dimca, Alexandru; Ilardi, Giovanna; Sticlaru, Gabriel

doi:10.1016/j.jalgebra.2022.10.019

Cited by 10 publications

(3 citation statements)

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“…) is called the exponents of the plane curve C. Some of the m−syzygy curves have been carefully studied and we refer [4] and [8] for the details. Let us recall here that…”

Section: Preliminariesmentioning

confidence: 99%

Construction of Free Curves by Adding Lines to a Given Curve

Dimca,

Ilardi,

Pokora

et al. 2023

Results Math

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In the present note we construct new families of free plane curves starting from a curve C and adding high order inflectional tangent lines of C, lines joining the singularities of the curve C, or lines in the tangent cone of some singularities of C. These lines L have in common that the intersection $$C \cap L$$ C ∩ L consists of a small number of points. We introduce the notion of a supersolvable plane curve and conjecture that such curves are always free, as in the known case of line arrangements. Some evidence for this conjecture is given as well, both in terms of a general result in the case of quasi homogeneous singularities and in terms of specific examples. We construct a new example of maximizing curve in degree 8 and the first and unique known example of maximizing curve in degree 9. In the final section, we use a stronger version of a result due to Schenck, Terao and Yoshinaga to construct families of free conic-line arrangements by adding lines to the conic-line arrangements of maximal Tjurina number recently classified by Beorchia and Miró-Roig in arXiv:2303.04665

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“…) is called the exponents of the plane curve C. Some of the m−syzygy curves have been carefully studied and we refer [4] and [8] for the details. Let us recall here that…”

Section: Preliminariesmentioning

confidence: 99%

Construction of Free Curves by Adding Lines to a Given Curve

Dimca,

Ilardi,

Pokora

et al. 2023

Results Math

Self Cite

View full text Add to dashboard Cite

show abstract

“…Our selection of singularities is well-motivated by works that have been published recently, see [6] as our main inspiration. Of course, all prescribed singularities in Assumption 1.5 are quasi-homogeneous since these are ADE singularities.…”

Section: Introductionmentioning

confidence: 99%

“…If we focus on the case when k = 2, then we should have to an arrangement of two conics such that its total Tjurina number is equal to τ (C) = 7. Based on a precise discussion from [6] we know that there is a unique up to the projective equivalence arrangement of two conics intersecting at a single point of type A 7 .…”

Section: Introductionmentioning

confidence: 99%