Understanding Dyson’s Lemma

In 1989, I proved a Dyson lemma for products of two smooth projective curves of arbitrary genus. In 1995, M. Nakamaye extended this to a result for a product of an arbitrary number of smooth projective curves of arbitrary genus, in a formulation involving an additional “perturbation divisor.” In 1998, he also found an example in which a hoped-for Dyson lemma is false without such a perturbation divisor. This talk will present some recent work suggesting that it may be possible to eliminate the perturbation divisor by using a different definition of “volume” at the points under consideration. Vague Definitions and History Let 0 ̸= P ∈ C[x1, x2] be of degree d1 in x1 and d2 in x2 ( d1 ≫ d2 ), and let Q1, . . . , Qs be points in C with distinct x1 coordinates and distinct x2 coordinates. Then Dyson’s lemma says that s ∑ i=1 VolP,d1,d2(Qi) ≤ 1 +O(d2/d1) . History: Theorem (Roth). Let α ∈ Q , let ε > 0 , and let C ∈ R . Then there are only finitely many p/q ∈ Q ( p, q ∈ Z , gcd(p, q) = 1 ) such that ∣∣∣pq − α ∣∣∣∣ ≤ C |q|2+ε . 1909 Thue d2 + 1 + ε 1921 Siegel min { d s+1 + s : 0 ≤ s < d } + ε 1947 Dyson √ 2d+ ε 1952 Gel’fond √ 2d+ ε? 1955 Roth 2 + ε Detailed Description of Dyson’s Lemma Let C1, . . . , Cn be smooth projective curves over C , let Y be an effective divisor on C1 × · · · × Cn , and let di = (Y . C̃i) for all i , where C̃i is a fiber of the map C1 × · · · × Cn → ∏ j ̸=i Cj . Assume that di > 0 for all i . Definition. For P ∈ C1 × · · · × Cn define the index of Y at P relative to d = (d1, . . . , dn) as td,Y (P ) = min { i1 d1 + · · ·+ in dn : ( ∂ ∂z1 )i1 · · · ( ∂ ∂zn )in f(P ) ̸= 0 } , where f is a local defining equation for Y at P and zi are local coordinates on Ci . We also define Vol(t) as Vol(t) = volume of { (x1, . . . , xn) ∈ [0, 1] : ∑ xi ≤ t } . Question. Given C1, . . . , Cn, Y, d1, . . . , dn as above, and points P1, . . . , Ps ∈ ∏ Ci lying in distinct fibers over Ci for all i , can one show that