Conductor and separating degrees for sets of points in $\mathbb{P}^r$ and in $\mathbb{P}^1 \times \mathbb{P}^1$

Marino, Lucia

Conductor and separating degrees for sets of points in

\mathbb{P}^{r}

and in

\mathbb{P}^{1}\times\mathbb{P}^{1}

Marino, Lucia

Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006), p. 397-421 / Harvested from Biblioteca Digitale Italiana di Matematica

Access to full text
Full (PDF)
Full (DjVu)

Abstract
Résumé

We attempt to generalize conductor degree's results, known in $\mathbb{P}^{2}$ , to the case of 0-dimensional schemes of $\mathbb{P}^{r}$ . In the first part of this paper, we consider the problem of characterizing the sequences generators's degrees of the conductor which are compatible with a fixed postulation (or Hilbert function) for a set of points in $\mathbb{P}^{r}$ and we determine the conductor degree of every point in a $r$ -partial intersection. In addition, we define the separating degree of a point for a 0-dimensional subscheme of a smooth quadric $Q=\mathbb{P}^{1}\times\mathbb{P}^{1}$ and we give some results in case of special subschemes.

In questo lavoro generalizziamo alcuni risultati sui gradi del conduttore, noti in $\mathbb{P}^{2}$ , al caso di schemi 0-dimensionali di $\mathbb{P}^{r}$ . Nella prima parte consideriamo il problema di caratterizzare la sequenza dei gradi dei generatori del conduttore in accordo con una fissata funzione di Hilbert per un insieme di punti in $\mathbb{P}^{r}$ e determiniamo il grado del conduttore di ogni punto in una $r$ -parziale intersezione. Inoltre diamo la definizione di separating degree di un punto per uno schema 0dimensionale su una quadrica liscia $Q=\mathbb{P}^{1}\times\mathbb{P}^{1}$ e proviamo dei risultati per esso nel caso di sottoschemi speciali.

Publié le : 2006-06-01

MR 2233144 | Zbl 1178.13007

@article{BUMI_2006_8_9B_2_397_0,
     author = {Lucia Marino},
     title = {Conductor and separating degrees for sets of points in $\mathbb{P}^r$ and in $\mathbb{P}^1 \times \mathbb{P}^1$},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {9-A},
     year = {2006},
     pages = {397-421},
     zbl = {1178.13007},
     mrnumber = {2233144},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2006_8_9B_2_397_0}
}

Marino, Lucia. Conductor and separating degrees for sets of points in $\mathbb{P}^r$ and in $\mathbb{P}^1 \times \mathbb{P}^1$. Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006) pp. 397-421. http://gdmltest.u-ga.fr/item/BUMI_2006_8_9B_2_397_0/

Bibliographie

[1] Abrescia, S. - Bazzotti, L. - Marino, L., Conductor Degree and Socle Degree, Le Matematiche, vol LVI- Fasc. I (2001), 129-148.

[2] Bazzotti, L., Doctoral Thesis, Florence, 2003.

[3] Geramita, A. V. - Kreuzer, M. - Robbiano, L., Cayley-Bacharach Schemes and their canonical modules, Transactions of the American Mathematical Society, vol 339 (1) (1993), 163-189. | Zbl 0793.14002

[4] Geramita, A. V. - Maroscia, P. - Roberts, L. G., The Hilbert function of a reduced k-algebra, J. Lond. Math. Soc. (2), 28 (1983), 443-452. | Zbl 0535.13012

[5] Giuffrida, S. - Maggioni, R. - Ragusa, A., On the postulation of 0-dimensional subschemes on a smooth quadric, Pacific J. of Mathematics, 155 (1992), 251-282. | Zbl 0723.14035

[6] Geramita, A.V. - Tadahito, H. - Yong Su, S., An alternative to the Hilbert Function for the Ideal of a Finite Set of Points in $\mathbb{P}^{n}$ , to appear: Illinois Journ. of Math (28 pages). | Zbl 0943.13012

[7] Maggioni, R. - Ragusa, A., A classification of arithmetically Cohen Macaulay varieties with given Hilbert Function, unpublished. | Zbl 0616.14026

[8] Migliore, J. C., Introduction to Liaison Theory and Deficiency Modules, Birkhäuser, 1998. | Zbl 0921.14033

[9] Orecchia, F., Points in generic positon and conductors of curves with ordinary singularities, J. Lond. Math. Soc. (2), 24 (1981), 85-96. | Zbl 0492.14017

[10] Peeva, I., Personal communications.

[11] Ragusa, A. - Zappala, G., Partial intersection and graded Betti numbers, to appear on Beiträge zur Algebra und Geometrie. | Zbl 1033.13004

[12] Sabourin, L., N-type vectors and the Cayley-Bacharach property, Comm. in Alg.30 (8) (2002), 3891-3915. | Zbl 1095.13526