Mathematics · the experiment
Find Nagata's frontier
Points scattered at random in the plane, a curve that must pass three times through each. Raise its degree up to the frontier set by Nagata's conjecture, then test its limits.
1 · Block the curve at sixteen points
Beginner
For r ≥ 10 very general points, what does Nagata's conjecture say about a curve of degree d passing m times through each?
Sixteen very general points, each passed through three times. Find the smallest degree of a curve that manages it.
2 · Count the conditions at ten points
Passing m times through a point imposes m(m+1)/2 conditions on the curve.
Intermediate
How many conditions do ten points impose, each passed through three times?
Ten very general points, each passed through three times. Find the smallest degree of a curve that manages it.
3 · Remove the tenth point
Nine points always lie on a cubic. Raised to the power m, that cubic becomes a curve of degree 3m passing m times through each of the nine points.
Expert
For r = 9, what does this curve do to the bound d > m√r?
4 · Look between the bound and the count
A hundred points, each passed through ten times: 5,500 conditions. The conjecture rules out every degree up to 10√100 = 100. The count guarantees a curve from degree 104, which has 5,565 coefficients; degree 103 has only 5,460.
Expert
With these two tools alone, what do we know about degrees 101, 102 and 103?
Frontier drawn
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