Resolutions of ideals of quasiuniform fat point subschemes of 𝐏²

Harbourne, Brian; Holay, Sandeep; Fitchett, Stephanie

doi:10.1090/S0002-9947-02-03124-0

Resolutions of ideals of quasiuniform fat point subschemes of $\mathbf P^2$
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Trans. Amer. Math. Soc. 355 (2003), 593-608 Request permission

Abstract:

The notion of a quasiuniform fat point subscheme $Z\subset \mathbf P^2$ is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal $I$ defining $Z$ 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 $m$th symbolic power $I(m;n)$ of an ideal defining $n$ general points of $\mathbf P^2$ when both $m$ and $n$ are large (in particular, for infinitely many $m$ for each of infinitely many $n$, and for infinitely many $n$ for every $m>2$). 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 $k$. As an incidental result, a bound for the regularity of $I(m;n)$ is given which is often a significant improvement on previously known bounds.

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