The regularity of points in multi-projective spaces

Abstract Let I=℘ 1 m 1 ∩⋯∩℘ s m s be the defining ideal of a scheme of fat points in P n 1 ×⋯× P n k with support in generic position. When all the m i 's are 1, we explicitly calculate the Castelnuovo–Mumford regularity of I . In general, if at least one m i ⩾2, we give an upper bound for the regularity of I , which extends a result of Catalisano, Trung and Valla.

[1]  A. Geramita,et al.  Ranks of tensors, secant varieties of Segre varieties and fat points , 2002 .

[2]  E. Guardo,et al.  Fat Points in ℙ1 × ℙ1 and Their Hilbert Functions , 2002, Canadian Journal of Mathematics.

[3]  Annetta Aramova,et al.  Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions , 2000 .

[4]  N. Trung,et al.  Upper Bounds for the Regularity Index of Fat Points , 1995 .

[6]  A. Gimigliano.  Regularity of Linear Systems of Plane Curves , 1989 .

[7]  S. Giuffrida,et al.  On the postulation of 0-dimensional subschemes on a smooth quadric , 1992 .

[8]  J. W. Hoffman,et al.  Regularity and Saturation in Biprojective Spaces , 2003, math/0305125.

[9]  J. William Hoffman,et al.  Castelnuovo–Mumford regularity in biprojective spaces , 2002 .

[10]  E. Guardo.  Fat points schemes on a smooth quadric , 2001 .

[11]  Diane Maclagan,et al.  Multigraded Castelnuovo-Mumford Regularity , 2003 .

[12]  M. V. Catalisano,et al.  A sharp bound for the regularity index of fat points in general position , 1993 .

[13]  The Defining Ideal of a Set of Points in Multi‐Projective Space , 2004, math/0409056.

[14]  Liaison and Related Topics: Notes from the Torino Workshop/School , 2002, math/0205161.

[15]  W. Bruns,et al.  Cohen-Macaulay rings , 1993 .

[16]  Aldo Conca,et al.  Castelnuovo-Mumford regularity of products of ideals , 2002 .

[17]  A. Tuyl.  The border of the Hilbert function of a set of points in Pn1×⋯×Pnk , 2002 .