Sur la caractéristique de la Jacobienne des systèmes linéaires de quadriques. (On the characteristic of the Jacobian of linear systems of quadrics). (French) Zbl 0621.14006

This paper, on classical algebraic geometry, appears as a list of definitions, theorems, proofs, results, without any explanation. It is not written the introdution, the purpose, the motivations, the main results of it. In reading the paper the impression of the reviewer is that to open a new book and to read from page 205 to page 223. From the bibliography one can understand that this paper is neither the first and, maybe, nor the last one of the series, but for a single paper it is better to have the begin and the end.
Passing now to review the content of the paper, the reviewer finds nice arguments and interesting results about characterizations of the rank of the Jacobian matrix of a linear system of quadrics (hypersurfaces) in the projective space \({\mathbb{P}}^ r\). It is not possible here to give a precise idea of the work, anyhow the results are of the following type. We call \(L_{d_ 1},L_{d_ 2},...,L_{d_ s}^ a \)chain of linear systems of quadrics if they are not reducible and if \(L_{d_ 1}\) and \(L_{d_ 2}\) have in common at least a quadric, their union system \(L_ a\) has in common with \(L_{d_ 3}\) at least a quadric and the system \(L_ b\) union of \(L_ a\) and \(L_{d_ 3}\) has in common with \(L_{d_ 4}\) at least a quadric, and so on. Then a linear system of quadrics \(L_ d\) containing a chain of linear systems of ”first kind” and no other quadric linearly independent, has the rank of the Jacobian matrix equal to r-k\(\leq d\) (k\(\geq 0)\) if and only if the quadrics of \(L_ d\), passing through a generic point of \({\mathbb{P}}^ r\), have in common a linear space \(S_{k+1}\).
Reviewer: E.Stagnaro


14C20 Divisors, linear systems, invertible sheaves
14N05 Projective techniques in algebraic geometry
Full Text: EuDML


[1] F. Palatini: Sulle superficie algebriche \(i\) cui \(S_h (h + 1)\) secanti non riempiono lo spazio ambiente. Atti R. Acc. Torino 41) · JFM 37.0667.01
[2] G. Bonferroni: Sui sistemi lineari di quadriche la cui jacobiana ha dimensione irregolare. R. Acc. Scienze Torino vol. 50) · JFM 45.1378.06
[3] A. Terracini: Alcune questioni sugli spazi tangenti e osculatori ad una varieta. Atti. R. Acc. Sc. Torino. Nota II, 51, (1916) III, 55)
[4] L. Muracchini: Sulle varietà \(V_5\) i cui spazi tangenti ricoprono una varietà \(W\) di dimensione inferiore all’ordinaria. (parte II) Riv. Mat. Univ. di Parma, 3, (1952), 75-89.
[5] S. Xambo: On projectives varieties of minimal degree. Collectanea Mathematica, Barcelona, vol. XXXII)
[6] L. Degoli: Un théorème sur les systèmes linéaires de quadriques à Jacobienne indéterminée. Studia Scientiarum Mathematicarum Hungarica Budapest. Tomo 17, (1982), 325-330. · Zbl 0555.51012
[7] L. Degoli: Due nuovi teoremi sui sistemi lineari di quadriche a Jacobiana identicamente nulla. Collectanea Mathematica, Barcelona, vol. XXXIII) · Zbl 0136.42504
[8] L. Degoli: Trois nouveaux théorèmes sur les systèmes linéaires de quadriques à Jacobienne identiquement nulle. Demonstratio Mathematica, Warszawa, Vol. 16) · Zbl 0542.14033
[9] L. Degoli: Alcuni teoremi sui sistemi lineari di quadriche a Jacobiana identicamente nulla. Revue d’analyse numérique et de théorie de l’approximation. - Cluj-Napoca. Tome 26 (49), (1984), 33-43.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.