Resolutions of ideals of quasiuniform fat point subschemes of

• Published in: Transactions of the American Mathematical Society 2003-02, Vol.355 (2), p.593-608
• Main Authors: Harbourne, Brian, Holay, Sandeep, Fitchett, Stephanie
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-02-03124-0

The notion of a quasiuniform fat point subscheme Z ⊂ P 2 Z\subset \mathbf P^2 is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal I I defining Z Z are put forward. In a large range of cases, it is shown that the Hilbert function conjecture implies the resolution conjecture. In addition, the main result gives the first determination of the resolution of the m m th symbolic power I ( m ; n ) I(m;n) of an ideal defining n n general points of P 2 \mathbf P^2 when both m m and n n are large (in particular, for infinitely many m m for each of infinitely many n n , and for infinitely many n n for every m > 2 m>2 ). Resolutions in other cases, such as “fat points with tails”, are also given. Except where an explicit exception is made, all results hold for an arbitrary algebraically closed field k k . As an incidental result, a bound for the regularity of I ( m ; n ) I(m;n) is given which is often a significant improvement on previously known bounds.