Preprint: On Nagata's Conjecture

Author: B. Harbourne
To appear, J. Alg.

Click here for pdf version of the paper.

Abstract: Given an integer m > 0 and given n general points of the projective plane, what is the least degree a(n,m) of a curve through the points having multiplicity at least m at each point? This preprint determines a new lower bound for a(n,m) and compares it to previously known bounds. Improved versions of the result are given in my survey article. Results even better than that (but only for characteristic 0) have now been worked out in a joint paper by J. Roe and me.

In More Detail: Related to his work on nonfinitely generated rings of invariants, Nagata computed a(n,m) for small values of n. If we define U(x) to be the least integer greater than or equal to x (i.e., U(x) means to round x Up) and D(x) to be the greatest integer function (round Down), it turns out that:

Fact: For n < 10, a(n,m) = U(cnm), where
c1 = c2 = 1, c3 = 1.5,
c4 = c5 = 2, c6 = 12/5,
c7 = 21/8, c8 = 48/17 and c9 = 3.

Nagata also posed the following conjecture (which he proved if n is a square bigger than 9):

Conjecture 1 (Nagata 1959): For n > 9, a(n,m) > mn1/2.

Two well known lower bounds for a(n,m) are derived by Bezout's Theorem (that the number of points, counted with multiplicity, at which two curves meet is equal to the product of their degrees):

Fact: Let n and m be any positive integers. Then: Roe Bound: In 1998, J. Roe developed an algorithm for computing an integer R(n,m) such that a(n,m) >= R(n,m). Via a sequence of specializations to infinitely near points, Roe shows that any curve of degree d passing through n > 2 points with multiplicity at least m at each point specializes to a curve passing through a single point with multiplicity R(n,m); hence 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.)

New Bound: Using a single specialization inspired by those of Roe, this preprint first shows:

Lemma (Harbourne 1999): Let d, r, m and n be positive integers such that dn1/2 <= r <= n; then a(n,m) >= U(mnd/r).

Taking the largest possible d (i.e., d = D(n1/2)) and then the least possible r (i.e., r = U(dn1/2)) now gives:

Theorem: Let m and n be positive integers; then a(n,m) >= U(mnD(n1/2)/U(D(n1/2)n1/2)).

It can then be checked that:

Fact: Let n be a positive integer. Then: Additional Background: It is believed that n > 9 general points of multiplicity m should impose independent conditions on forms of degree d, as long as the dimension (d2 + 3d + 2)/2 of the space of forms of degree d is at least the number of expected conidtions, n(m2 + m)/2. In any case, (d2 + 3d)/2 - n(m2 + m)/2 is a lower bound for the projective dimension of the space of curves through n points with multiplicity at least m at each point. This leads to the following fact and conjecture:

Fact: For all positive n and m we have a(n,m) <= D([1/4 + n(m2 + m)]1/2) - 1/2.

Conjecture 2: For n > 9, a(n,m) = D([1/4 + n(m2 + m)]1/2) - 1/2.

Here is what seems to be known about the actual value of a(n,m): Here is a web form for computing the various bounds mentioned above.
Enter the number n of points here:
Enter the uniform multiplicity m at each point here: