Problems and Progress on Points in P2

Speaker: B. Harbourne
Affiliation: Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, NE 68588-0323, USA

email: bharbour@math.unl.edu
web: http://www.math.unl.edu/~bharbour/

Conference: Workshop on Zero-Dimensional Schemes and Applications

This abstract has been revised and updated to include results known to the author as of March 14, 2000.

Notations

Major Problems

Determine hIZ(m)

For t sufficiently small, we know hIZ(m)(t) = 0, while for t sufficiently large we know hIZ(m)(t) = [(t+2)(t+1) - m1(m1+1) - ... - mn(mn+1)]/2. We regard max(0, [(t+2)(t+1) - m1(m1+1) - ... - mn(mn+1)]/2) as the ``expected'' value of hIZ(m)(t), although there are cases for which this expected value is known not to be the actual value; when hIZ(m)(t) = [(t+2)(t+1) - m1(m1+1) - ... - mn(mn+1)]/2 we say that Z(m) imposes independent conditions on forms of degree t. Short of determining hIZ(m)(t) for every t, it is also of interest to bound a(m) below and to bound tind(m) above (where a(m) is defined above and tind(m) is the least degree t such that Z(m) imposes independent conditions on forms of degree t). As with a(m) and a(n,m), we will also write tind(n,m) for tind(m) when m1 = ... = mn.

For n <= 9, hIZ(m) is known (see M. Nagata, On rational surfaces, II. Mem. Coll. Sci. Kyoto (A) 33 (1960), 271-293, or, for a more modern and more general approach, see B. Harbourne, Anticanonical Rational Surfaces, Trans. Amer. Math. Soc. 349 (1997), 1191-1208). Thus interest has centered on the case that n > 9.

[P1] The Segre-Harbourne-Gimigliano-Hirschowitz [SHGH] Conjecture

Several people have put forward conjectures regarding the value of hIZ(m): These conjectures all turn out to be equivalent, and allow one to explicitly compute all values of hIZ(m) (the equivalence of Segre's conjecture to the others was only recently recognized, by Ciliberto, Clemens and Miranda). Here are some additional references:

Rather than discuss the general form of the SHGH conjecture, we will consider a special case. Say that m and by extension Z(m) is quasi-uniform if n > 9 and m1 = ... = m9 >= m10 >= ... >= mn >= 0, and that Z(m) is uniform if n > 9 and m1 = ... = mn. Then the conjectures above imply:

(Quasi-)Uniform SHGH Conjecture: If Z(m) is (quasi-)uniform, then hIZ(m) has its expected value in every degree t >= 0.

Some results are known:
Here is a web form for computing actual and expected values of the Hilbert function.
Enter the number n of points here:
Enter a single multiplicity m here:

Bounding a(n,m)

There is also a lot of work on bounding a(n,m).

Easy Fact: If n >= 0 and m >= 0, then a(n,m) <= D([1/4 + n(m2 + m)]1/2 - 1/2). (This follows from Riemann-Roch. Moreover, for n > 9, the SHGH Conjecture implies this is an equality.)

Nagata's Conjecture (M. Nagata, On the 14th Problem of Hilbert, Amer. J. Math. 33 (1959), 766-772): If n > 9 with m > 0, then a(n,m) > mn1/2.

Nagata proved this Conjecture for any square n > 9. The strict inequality is a bit tricky; an easy argument specializing the n points to a curve of degree n1/2 gives a(n,m) >= mn1/2 when n is a square. More generally, specializing to a curve of degree D(n1/2) or, resp., degree U(n1/2) gives the bounds: The best general bound (applicable in all characteristics, almost always better and never worse than mD(n1/2) or mn/U(n1/2)), seems to be (see B. Harbourne, On Nagata's Conjecture, preprint (1999)): In general, it does not seem easy to tell which d and r give the best result, but taking d = D(n1/2) and r = U(D(n1/2)n1/2) is quite good and gives: However, J. Roé (1998) defines a quantity R(n,m) for n > 2 and any m such that a(n,m) >= R(n,m). For m not bigger than about n1/2, R(n,m) seems overall to be the best lower bound constructed so far since, for m < n1/2, R(n,m) seems (although this does not seem easy to prove) to exceed Nagata's conjectured bound; here is a link to a more complete description of how R(n,m) is defined and computed. See B. Harbourne, On Nagata's Conjecture, preprint (1999), for discussion and references, or link here for a more complete discussion. (Also worth mentioning are bounds given by G. Xu for integral curves in characteristic 0; see ``Curves in P2 and symplectic packings,'' Math. Ann. 299 (1994), 609-613.)
Here is a web form for computing some of the bounds on a(n,m) mentioned above.
Enter the number n of points here:
Enter a single multiplicity m here:

Bounding tind(n,m)

A connection between bounding tind(n,m) and a(n,m) has been noted by Ziv Ran: if a(n,m) >= cm for some constant c, then tind <= max(n1/2, n/c)(m+1) - 2. (If c < n1/2, this simplifies to tind <= U(n(m+1)/c) - 3.)

Thus the bounds above on a(m,n) give bounds on tind(n,m). In particular, gives what seem to be the best general bound for large m: When n is a square, one can do better, however:
Here are some previous bounds for tind(n,m): When n is a square and m is large enough, [3] is always the best bound, since it is exact. When m is small (m up to about n1/2 when n is not a square), numerical evidence suggests [2] d1(m,n) is the best bound, although [4] is sometimes close. When m is larger than about n1/2 and n is not a square, [1] seems to be the best bound.
Here is a web form for computing some of the bounds on tind(n,m) mentioned above.
Enter the number n >= 3 of points here:
Enter a single multiplicity m >=1 here:

[P2] Determine F0(Z(m)) as a graded R-module

Notes: Question: What should one expect for the structure of F0(Z(m)) for arbitrary m (i.e., when the multiplicities are not all equal)?