We define an algebraic analogue, in the case of jacobians of curves, of the height jump divisor introduced recently by R. Hain. We give explicit combinatorial formulae for the height jump for families of semistable curves using labelled reduction graphs. With these techniques we prove a conjecture of Hain on the effectivity of the height jump, and also give a new proof of a theorem of Tate, Silverman and Green on the variation of heights in families of abelian varieties.

A discriminant algebra operation sends a commutative ring R and an R-algebra A of rank n to an R-algebra ∆ A/R of rank 2 with the same discriminant bilinear form. Constructions of discriminant algebra operations have been put forward by Rost, Deligne, and Loos. We present a simpler and more explicit construction that does not break down into cases based on the parity of n. We then prove properties of this construction, and compute some examples explicitly.

To generalize the notion of Galois closure for separable field extensions, we devise a notion of G-closure for algebras of commutative rings R → A, where A is locally free of rank n as an R-module and G is a subgroup of S n . A G-closure datum for A over R is an R-algebra homomorphism ϕ : (A ⊗n ) G → R satisfying certain properties, and we associate to a closure datum ϕ a closure algebra A ⊗n ⊗ (A ⊗n ) G R. This construction reproduces the normal closure of a finite separable field extension if G is the corresponding Galois group. We describe G-closure data and algebras of finite étale algebras over a general connected ring R in terms of the corresponding finite sets with continuous actions by the étale fundamental group of R. We show that if 2 is invertible, then A n -closure data for free extensions correspond to square roots of the discriminant, and that D 4 -closure data for quartic monogenic extensions correspond to roots of the cubic resolvent. This is an updated and revised version of the author's Ph.D. thesis.

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