2014
DOI: 10.5802/jep.10
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Syzygies and logarithmic vector fields along plane curves

Abstract: For a reduced curve C : f = 0 in the complex projective plane P 2 , we study the set of jumping lines for the rank two vector bundle T C on P 2 , whose sections are the logarithmic vector fields along C. We point out the relations of these jumping lines with the Lefschetz type properties of the Jacobian module of f and with the Bourbaki ideal of the module of Jacobian syzygies of f . In particular, when the vector bundle T C is unstable, a line is a jumping line if and only if it meets the 0-dimensional subsch… Show more

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Cited by 49 publications
(80 citation statements)
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“…Our interest in this type of results comes from the fact that hypersurfaces with isolated weighted homogeneous singularities have been recently shown to possess deep properties with respect to the syzygies of their Jacobian ideals, see [7], [8], [10], [12]. In order to extend such properties to the non weighted homogeneous singularities, one has perhaps to understand first better the relations between syzygies and weighted homogenenous singularities.…”
Section: Introduction and Statement Of Resultsmentioning
confidence: 99%
“…Our interest in this type of results comes from the fact that hypersurfaces with isolated weighted homogeneous singularities have been recently shown to possess deep properties with respect to the syzygies of their Jacobian ideals, see [7], [8], [10], [12]. In order to extend such properties to the non weighted homogeneous singularities, one has perhaps to understand first better the relations between syzygies and weighted homogenenous singularities.…”
Section: Introduction and Statement Of Resultsmentioning
confidence: 99%
“…Let If denote the saturation of the ideal Jf with respect to the maximal ideal m=(x,y,z) in S and consider the local cohomology group, usually called the Jacobian module of f , Nfalse(ffalse)=If/Jf=Hboldm0false(M(f)false).The graded S ‐module ARfalse(ffalse)=ARfalse(Cfalse)S3 of all Jacobian relations of f is defined by AR(f)k:={false(a,b,cfalse)Sk3afx+bfy+cfz=0}.Its sheafification EC:=AR(f) is a rank two vector bundle on P2, see for details. More precisely, one has EC=Tfalse⟨Cfalse⟩false(1false), where TC is the sheaf of logarithmic vector fields along C as considered for instance in . We set ar(f)m=dimAR(f)m=dimH0(double-struckP2,ECfalse(m...…”
Section: Various Prerequisitesmentioning
confidence: 99%
“…In this section we look at the dimensions h0((scriptJffalse|Ctrue)(d)true)=prefixdimH0true(true(Jffalse|C)false(dfalse)) and h1((scriptJffalse|Ctrue)(d)true)=prefixdimH1true(true(Jffalse|C)false(dfalse)), using results by Sernesi in and our study of the Jacobian syzygies and Jacobian module N(f) in .…”
Section: Jacobian Syzygies Jacobian Module and Equianalytic Deformatmentioning
confidence: 99%
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