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.

- R = k[
**P**^{2}] = homogeneous coordinate ring of**P**^{2}= polynomial ring in three indeterminates over algebraically closed field k - p
_{1}, ... , p_{n}are general points in**P**^{2} -
**m**= (m_{1}, ... , m_{n}) is a vector of nonnegative integer multiplicities - I
_{pi}is the ideal in R generated by all forms vanishing at p_{i} - Z(
**m**) denotes the fat points subscheme m_{1}p_{1}+ ... + m_{n}p_{n} - Z(n,m) denotes the fat points subscheme
m(p
_{1}+ ... +p_{n}) - The defining ideal for Z(
**m**) is the intersection I_{Z(m)}in R of the ideals I_{p1}^{m1}, ... , I_{pn}^{mn}(where I_{pi}^{mi}is the m_{i}th power of the ideal I_{pi}) - 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 (I_{Z(m)})_{t}of I_{Z(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 - h
_{IZ(m)}is the Hilbert function of I_{Z(m)}; i.e., h_{IZ(m)}(t) = dim_{k}(I_{Z(m)})_{t} - 0 -> F
_{1}(Z(**m**)) -> F_{0}(Z(**m**)) -> I_{Z(m)}-> 0 denotes a minimal free graded resolution of I_{Z(m)}over R

- [P1] Determine h
_{IZ(m)}.- In what degrees t is (I
_{Z(m)})_{t}=0? - In what degrees t does Z(
**m**) impose independent conditions on forms of degree t?

- In what degrees t is (I
- [P2] Determine F
_{0}(Z(**m**)) as a graded R-module.- What would be a reasonable conjecture for the structure of
F
_{0}(Z(**m**)) as a graded R-module?

- What would be a reasonable conjecture for the structure of
F

For n <= 9, h

- 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 gé
né riques, Journ. Reine Angew. Math. 397 (1989), 208-213.

- B. Harbourne, Points in Good Position in
**P**^{2}, 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

(Quasi-)Uniform SHGH Conjecture: If Z(

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 ré
union de 4 ^{h}gros points géné riques de P ^{2}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).

- (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 ré
- 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.

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) > mn

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 n

- a(n,m) >= mD(n
^{1/2}) - a(n,m) >= mn/U(n
^{1/2})

- a(n,m) >= mnd/r, whenever (r/d)
^{2}>= n >= r- For the best choice of r and d we always
have n
^{1/2}- 1/(2n^{3/2}) > nd/r >= n^{1/2}- 1/n^{1/2} - For n = x
^{2}+2, the best possible choices of r and d are r = x^{2}+1, and d = x, which puts nd/r quite close to the bound n^{1/2}- 1/(2n^{3/2}).

- For the best choice of r and d we always
have n

- a(n,m) >= U(mnD(n
^{1/2})/U(D(n^{1/2})n^{1/2}))- Note that nD(n
^{1/2})/U(D(n^{1/2})n^{1/2}) >= n^{1/2}- 1/n^{1/2}, but that when n - (D(n^{1/2}))^{2}is even, then nD(n^{1/2})/U(D(n^{1/2})n^{1/2}) >= n^{1/2}- 1/(2n^{1/2}).

- Note that nD(n

Here is a web form for computing some of the bounds on a(n,m) mentioned above.

Thus the bounds above on a(m,n) give bounds on t

- a(n,m) >= mnd/r, whenever (r/d)
^{2}>= n >= r

- [1] t
_{ind}(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(n
^{1/2}+ 1/(2n^{3/2})) < mr/d <= m(n^{1/2}+ 1/n^{1/2})

- ignoring all terms but that involving m, this bound
(taking best choices for r and d) satisfies
m(n
- [2] Also see the bound d
_{1}(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).

- [3] t
_{ind}(n,m) <= mU(n^{1/2}) + U((U(n^{1/2}) - 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 t
_{ind}(n,m) cannot be less than mU(n^{1/2}) + U((U(n^{1/2}) - 3)/2), so in fact t_{ind}(n,m) = mU(n^{1/2}) + U((U(n^{1/2}) - 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(n
^{1/2}) + U((U(n^{1/2}) - 3)/2) is equal to the expected value D([1/4 + n(m^{2}+ m)]^{1/2}- 1/2) of a(n,m).

Here are some previous bounds for t

- 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
gé
né riques, Journ. Reine Angew. Math. 397 (1989), 208-213) give upper bounds of approximately m(2n) ^{1/2}for t_{ind}(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 t_{ind}(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 t_{ind}(n,m)

Here is a web form for computing some of the bounds on t

- F
_{0}(Z(**m**)) and h_{IZ(m)}together determine F_{1}(Z(**m**)). - F
_{0}(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 I_{Z(m)}is

0 -> R[-a-2] ^{a+1-h}x R[-a-1]^{c}-> R[-a-1]^{b}x R[-a]^{h}-> I_{Z(m)}-> 0,

where a = a(**m**), h = h_{IZ(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 >= (n
^{1/2}- 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 >= (n
^{1/2}- 2)/4, then the expected resolution given above holds whenever a(n,m) has its expected value of mn^{1/2}+ (n^{1/2}-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
t
_{ind}<= 4m + 1 by a bound given above), and, in characteristic 0, (applying Evain's paper mentioned above) for m >= (n^{1/2}- 2)/4 if n > 9 is an even square not divisible by a prime bigger than 5.

- No general conjecture for F
_{0}(Z(**m**)) has been given for n > 8.

- We know F
_{0}(Z(**m**)) is the direct sum over all degrees t of R[-t]^{ct}, where c_{t}is the dimension of the cokernel of the multiplication map R_{1}x (I_{Z(m)})_{t-1}-> (I_{Z(m)})_{t}. Let X be the surface obtained by blowing up the points p_{1}, ... , p_{n}, with L a general line on X and E_{1}, ... , E_{n}the exceptional divisors of the blow ups. Let F_{t}(**m**) = (tL - m_{1}E_{1}- ... - m_{n}E_{n}), where**m**= (m_{1}, ... , m_{n}). Then to compute c_{t}for all t for every**m**= (m_{1}, ... , m_{n}), it turns out to be enough to be able in general to compute c_{t}for those t such that F_{t}(**m**) meets every exceptional curve on X nonnegatively. Assuming the SHGH Conjecture, it is in turn enough in general just to compute c_{tind}. There are two nontrivial cases:- a(
**m**) < t_{ind}(**m**), and - a(
**m**) = t_{ind}(**m**).

- a(
- For the case a(
**m**) < t_{ind}(**m**), it is enough (again assuming SHGH, as shown in S. Fitchett's 1997 thesis) to consider**m**and t such that F_{t}(**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 m_{i}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 = E
_{9}+ A + bK_{X}where:- K
_{X}= 3_{0}L - E_{1}- ... - E_{9}, - A = a
_{0}L + a_{1}E_{1}+ ... + a_{8}E_{8}, - a
_{0}, ... , a_{8}are any integers such that 3a_{0}- a_{1}- ... - a_{8}= 0 (i.e., A.K_{X}= 0), and - b = (a
_{0}^{2}- a_{1}^{2}- ... - a_{8}^{2})/2 (i.e., b = A^{2}/2).

_{1}- E_{2}). Then with E = E_{9}+ A + bK_{X}and with a^{2}+ a < s < 2a^{2}- a, this gives F_{t}(**m**) = L + sE satisfying u < s < U but for which it is not known what value to expect for c_{t}. - K

- Here
is a link to a web based computation of the
minimal free resolution of I
_{Z(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.