Preprint: On Nagata's Conjecture
Author: B. Harbourne
To appear, J. Alg.
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
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),
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
Fact: Let n and m be any positive integers. Then:
Roe Bound: In 1998,
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.)
- a(n,m) >= mD(n1/2), and
- a(n,m) >= mn/U(n1/2).
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:
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:
- nD(n1/2)/U(D(n1/2)n1/2) >=
D(n1/2), with equality if and only if n or n-1 is a square;
- nD(n1/2)/U(D(n1/2)n1/2) >=
n/U(n1/2), with equality if and only if n, n+1 or n+2
is a square;
- limm->\inf nD(n1/2)/U(D(n1/2)n1/2) -
R(n,m) = \inf, hence U(mnD(n1/2)/U(D(n1/2)n1/2))
> R(n,m) for all m sufficiently large.
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.