Nagata’s conjecture and maximal Seshadri constants

The explanation

Pick nine points in the plane, scattered at random. Is there a cubic curve through all nine? A cubic equation has 10 coefficients. Passing through a point is one linear condition on them. Nine conditions on 10 unknowns always leave a nonzero solution. So a cubic through the nine points always exists. Add a tenth random point and the count becomes 10 conditions on 10 unknowns. Now the cubic usually disappears.

Nagata's conjecture turns this into a rule. Take a plane curve of degree d, meaning its equation has degree d. Its multiplicity at a point counts how many of its branches cross there. A simple pass counts 1. A curve that crosses itself three times there counts 3. The rule says: at r random points, with r at least 10, add up the multiplicities. The total must stay below d times the square root of r. For the tenth-point cubic, the total would be 10. The limit is 3 times √10, about 9.49. So the cubic is forbidden. At nine points the limit is exactly 3 times 3, which is 9. The cubic sits exactly on the line, so the strict rule cannot start at nine.

Counting suggests the square root. A degree-d equation has (d+1)(d+2)/2 coefficients. Multiplicity m at r points costs at most r·m(m+1)/2 conditions. For large curves the two counts balance when d equals m times √r. But counting alone proves nothing below that line. Conditions can overlap, and then fewer of them really bite. The paper names the hard part: the dependencies among the conditions must be controlled for every mSource 1.

Masayoshi Nagata posed the problem in 1959, in his work on Hilbert's fourteenth problem. He proved the rule when r is a perfect square, at least 16. Every other count stayed open. Joaquim Roé proved a general bound in 2001. With all multiplicities equal to m and r at least 10, d exceeds m(√(r−1) − π/8). At ten points that gives about 2.61 times m, against the target of about 3.162. Later work by Thomas Eckl and by Ciliberto, Dumitrescu, Miranda and Roé pushed the ten-point bound to 117/37. That is about 3.16216. The target √10 is about 3.16228.

The gap looks tiny, but it hides real curves. Take ten random points, each crossed 37 times. The old bound allows degree 117, since 117/37 just meets it. Nagata's rule needs degree above 37 times √10, about 117.004. So it demands at least 118. The count already leans against degree 117: 7021 coefficients against 7030 conditions. Overlaps could still let a curve slip through. Only a proof can close that door.

The first paper proves the full rule for every r from 10 upSource 2. It covers curves made of several pieces, repeated pieces, and unequal multiplicities at different pointsSource 3. Here very general means one thing precisely. A thin set of bad point positions is removed once, for all curves at the same timeSource 4. It is a countable union of lower-dimensional pieces, so random points avoid it. The paper also extends the result to other number systems that behave like the complex numbersSource 5.

The other papers move from the plane to other shapes. A Seshadri constant measures how far a curve's degree must outweigh its multiplicities at r chosen points. There is a simple ceiling for it, set by the shape's size and by rSource 6. The second paper covers any smooth surface with any chosen notion of degree. It proves that random points reach that ceiling once r is large enoughSource 7. How large depends on the surface, and no explicit number is givenSource 8. Two later papers claim to prove the same in every higher dimension. One works over the complex numbersSource 9. The other works in positive characteristic, number systems where adding some prime number of 1s gives 0Source 10.

How sure is this? A computer proof checker called Lean has checked the plane rule for every r from 10. That check covers reducible curves, repeated pieces and unequal multiplicities. Lean has also checked the surface result. Its threshold is not explicit. The higher-dimensional and positive-characteristic papers are outside the Lean scope note. No specialists have published a review yet.

What it doesn't show. This does not give an explicit number of points beyond which the surface and higher-dimensional results start.