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

• R = k[P2] = homogeneous coordinate ring of P2 = polynomial ring in three indeterminates over algebraically closed field k
• p1, ... , pn are general points in P2
• m = (m1, ... , mn) is a vector of nonnegative integer multiplicities
• Ipi is the ideal in R generated by all forms vanishing at pi
• Z(m) denotes the fat points subscheme m1p1 + ... + mnpn
• Z(n,m) denotes the fat points subscheme m(p1 + ... +pn)
• The defining ideal for Z(m) is the intersection IZ(m) in R of the ideals Ip1m1, ... , Ipnmn (where Ipimi is the mith power of the ideal Ipi)
• D(x) = the greatest integer less than or equal to x (i.e., round x Down)
• U(x) = the least integer greater than or equal to x (i.e., round x Up)
• a(m) = the least degree t such that the homogeneous component (IZ(m))t of IZ(m) is nontrivial in degree t
• Alternatively, if m and n are positive integers, then a(n,m) will denote a(m), where m = (m, ... , m) has n entries
• hIZ(m) is the Hilbert function of IZ(m); i.e., hIZ(m)(t) = dimk(IZ(m))t
• 0 -> F1(Z(m)) -> F0(Z(m)) -> IZ(m) -> 0 denotes a minimal free graded resolution of IZ(m) over R

Major Problems

• [P1] Determine hIZ(m).
• In what degrees t is (IZ(m))t=0?
• In what degrees t does Z(m) impose independent conditions on forms of degree t?
• [P2] Determine F0(Z(m)) as a graded R-module.
• What would be a reasonable conjecture for the structure of F0(Z(m)) as a graded R-module?

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):
• B. Segre: Alcune questioni su insiemi finiti di punti in Geometria Algebrica, Atti del Convegno Internaz. di Geom. Alg., Torino (1961).
• B. Harbourne: The geometry of rational surfaces and Hilbert functions of points in the plane, Can. Math. Soc. Conf. Proc. 6 (1986), 95-111.
• A. Gimigliano: On linear systems of plane curves. Ph.D. thesis, Queen's University (1987).
• A. Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles gériques, Journ. Reine Angew. Math. 397 (1989), 208-213.
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:
• B. Harbourne, Points in Good Position in P2, in: Zero-dimensional schemes, Proceedings of the International Conference held in Ravello, Italy, June 8-13, 1992, De Gruyter, 1994.
• R. Miranda, Linear Systems of Plane Curves, Notices Amer. Math. Soc., Feb. 1999 (abstract pdf(146K)).

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:
• The Uniform SHGH Conjecture holds when:
• (in characteristic 0) n is a square not divisible by a prime bigger than 5. See L. Evain's paper ``La fonction de Hilbert de la réunion de 4h gros points gériques de P2 de même multiplicité'', J. Alg. Geom. 8 (1999), 787--796.
• (in characteristic 0) m <= 12: see C. Ciliberto/R. Miranda, Linear Systems of plane curves with base points of equal multiplicity, preprint (1998), to appear, Trans. Amer. Math. Soc.
• n is sufficiently large compared with m: see J. Alexander and A. Hirschowitz, "An Asympotic Vanishing Theorem for Generic Unions of Multiple Points", preprint (1997).
• Various results when no multiplicity is more than 4 have been obtained for the full SHGH Conjecture.

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:
• a(n,m) >= mD(n1/2)
• a(n,m) >= mn/U(n1/2)
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)):
• a(n,m) >= mnd/r, whenever (r/d)2 >= n >= r
• For the best choice of r and d we always have n1/2 - 1/(2n3/2) > nd/r >= n1/2 - 1/n1/2
• For n = x2+2, the best possible choices of r and d are r = x2+1, and d = x, which puts nd/r quite close to the bound n1/2 - 1/(2n3/2).
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:
• a(n,m) >= U(mnD(n1/2)/U(D(n1/2)n1/2))
• Note that nD(n1/2)/U(D(n1/2)n1/2) >= n1/2 - 1/n1/2, but that when n - (D(n1/2))2 is even, then nD(n1/2)/U(D(n1/2)n1/2) >= n1/2 - 1/(2n1/2).
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,
• a(n,m) >= mnd/r, whenever (r/d)2 >= n >= r
gives what seem to be the best general bound for large m:
• [1] tind(n,m) <= U((m+1)r/d) - 3, whenever (r/d)2 >= n >= r
• ignoring all terms but that involving m, this bound (taking best choices for r and d) satisfies m(n1/2 + 1/(2n3/2)) < mr/d <= m(n1/2 + 1/n1/2)
• [2] Also see the bound d1(m,n) given by J. Roé (Linear systems of plane curves with imposed multiple points, (2000) preprint), which, although hard to analyze, seems to be the best bound currently available when m is not too large (always at least as good as (m+1)((n + 2.5)1/2+Pi/8)-1).
When n is a square, one can do better, however:
• [3] tind(n,m) <= mU(n1/2) + U((U(n1/2) - 3)/2)
• See Lemma 5.3 of Harbourne/Holay/Fitchett HHF 1999.
• When n > 1 is a square and m is sufficiently large, it is easy to check that tind(n,m) cannot be less than mU(n1/2) + U((U(n1/2) - 3)/2), so in fact tind(n,m) = mU(n1/2) + U((U(n1/2) - 3)/2) for m >> 0 when n > 1 is a square.
• When n > 9 is a square and m is sufficiently large, it is interesting to note that mU(n1/2) + U((U(n1/2) - 3)/2) is equal to the expected value D([1/4 + n(m2 + m)]1/2 - 1/2) of a(n,m).

Here are some previous bounds for tind(n,m):
• A. Gimigliano (Regularity of Linear Systems of Plane Curves, J. Alg 124 (1989), 447 - 460) and A. Hirschowitz, (Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles gériques, Journ. Reine Angew. Math. 397 (1989), 208-213) give upper bounds of approximately m(2n)1/2 for tind(n,m)
• G. Xu, Ample line bundles on smooth surfaces, Journ. Reine Angew. Math. 469 (1995), 199 - 209 gives an upper bound of approximately m(10n/9)1/2 for tind(n,m)
• [4] E. Ballico (Curves of minimal degree with prescribed singularities, (1997), preprint) gives an upper bound of approximately m((n+1)1/2 + 1) 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:
• F0(Z(m)) and hIZ(m) together determine F1(Z(m)).
• F0(Z(m)) is now known for any m with n <= 8 (in preparation, Fitchett, Harbourne, and Holay).
• (Quasi-)Uniform Resolution Conjecture: (Harbourne 1997, HHF 1999) If n > 9 and m is (quasi-)uniform, then the minimal free resolution of IZ(m) is
0 -> R[-a-2]a+1-h x R[-a-1]c -> R[-a-1]b x R[-a]h -> IZ(m) -> 0,

where a = a(m), h = hIZ(m)(a), b = max(0,a+2-2h), and c = max(0,2h-a-2).
• Most of the evidence for this conjecture is for the uniform case:
• The Uniform Resolution Conjecture holds for m=1 (A. V. Geramita, D. Gregory, L. Roberts, Minimal ideals and points in projective space, J. Pure Appl. Alg. 40 (1986), 33-62).
• The Uniform Resolution Conjecture holds for m=2 (M. Idà, The minimal free resolution for the first infinitesimal neighborhoods of n general points in the plane, J. Alg. 216 (1999), 741-753).
• Theorem: The SHGH Conjecture implies the Uniform Resolution Conjecture for (HHF 1999):
• infinitely many m for each n > 9,
• for all m >= (n1/2 - 2)/4 if n is an even square, and
• for all m >= (n - 9)/8 if n is an odd square.
• In fact, if n > 9 is an even square and m >= (n1/2 - 2)/4, then the expected resolution given above holds whenever a(n,m) has its expected value of mn1/2 + (n1/2-2)/2 (see HHF 1999). Thus, for even squares, proving the Uniform Resolution Conjecture (for m sufficiently large) amounts to proving that a(m,n) has its expected value.
• Therefore the Uniform Resolution Conjecture holds for n = 16 (note that SHGH holds for n = 16 uniform multiplicities since a(16,m) >= 4m + 1 by Nagata and tind <= 4m + 1 by a bound given above), and, in characteristic 0, (applying Evain's paper mentioned above) for m >= (n1/2 - 2)/4 if n > 9 is an even square not divisible by a prime bigger than 5.
• No general conjecture for F0(Z(m)) has been given for n > 8.
Question: What should one expect for the structure of F0(Z(m)) for arbitrary m (i.e., when the multiplicities are not all equal)?
• We know F0(Z(m)) is the direct sum over all degrees t of R[-t]ct, where ct is the dimension of the cokernel of the multiplication map R1 x (IZ(m))t-1 -> (IZ(m))t. Let X be the surface obtained by blowing up the points p1, ... , pn, with L a general line on X and E1, ... , En the exceptional divisors of the blow ups. Let Ft(m) = (tL - m1E1 - ... - mnEn), where m = (m1, ... , mn). Then to compute ct for all t for every m = (m1, ... , mn), it turns out to be enough to be able in general to compute ct for those t such that Ft(m) meets every exceptional curve on X nonnegatively. Assuming the SHGH Conjecture, it is in turn enough in general just to compute ctind. There are two nontrivial cases:
• a(m) < tind(m), and
• a(m) = tind(m).
• For the case a(m) < tind(m), it is enough (again assuming SHGH, as shown in S. Fitchett's 1997 thesis) to consider m and t such that Ft(m) = L + sE for some exceptional curve E on X, with 1 <= s <= L.E and also with u < s < U, where u is the minimum of t and t - m, U is the maximum of t and t - m, and m is the maximum of mi over all i.
• It is easy to generate test cases, especially with n = 9. The exceptional curves are precisely the divisors of the form E = E9 + A + bKX where:
• KX = 30L - E1 - ... - E9,
• A = a0L + a1E1 + ... + a8E8,
• a0, ... , a8 are any integers such that 3a0 - a1 - ... - a8 = 0 (i.e., A.KX = 0), and
• b = (a02 - a12 - ... - a82)/2 (i.e., b = A2/2).
So for example, we can take A = a(E1 - E2). Then with E = E9 + A + bKX and with a2 + a < s < 2a2 - a, this gives Ft(m) = L + sE satisfying u < s < U but for which it is not known what value to expect for ct.

• Here is a link to a web based computation of the minimal free resolution of IZ(m) for n <= 7 and n = 8.
• Here is a link to a web based computation of the resolution (or at least the expected resolution) for the Uniform Resolution Conjecture.