One of the main results of the article is stated for general models of a smooth curve over a discrete valuation ring, where the models are allowed to admit a non-regular locus. The result computes the valuation of the Deligne discriminant, as defined in the original article. More precisely, the computation is in terms of the deviation from the possible expectation in terms of the Artin conductor, naively generalizing from the regular case.

All the components of the non-regular locus contribute to the deviation with, in principle, computable terms. However, the computation is only correctly performed whenever the total space X is normal. The modification to the non-normal case was mistakenly added at a late stage of the submission. The mistake as such appears, in [1, Proposition 4.2], where one would also have to take into account the normalization map. The author apologizes for this.

To correctly state the result, we recall the setting. We consider X → S a flat projective local complete intersection morphism with geometrically connected fibers of dimension one. Here S = Spec ⁡ R is supposed to be the spectrum of a discrete valuation ring with perfect residue field k ⁢ ( s ) and valuation v. Since we are interested in degenerations, we will assume the general fiber smooth.

In this setting, one defines the Artin conductor as the difference of the ℓ -adic Euler characteristics of the geometric generic, resp. special, fibers ( X η ¯ resp. X s ¯ ), with a correction term coming from the Swan conductor:

Here the Swan conductor is defined as in, e.g., [2]. The definition is invariant upon taking completions of R, by [2, Corollary 6.3.4].

Suppose now that X is normal. If x ∈ X is a non-normal point, necessarily lying over the special fiber, there exists a desingularization π : X ′ → X locally around x. Denote by E = π - 1 ⁢ ( x ) the exceptional divisor and by b i ⁢ ( E ) the ℓ -adic Betti numbers. Also, consider the coherent 𝒪 X -module R 1 ⁢ π ∗ ⁢ 𝒪 X ′ . One denotes by p g the length of this as an R-module. Finally, if Γ denotes the discrepancy divisor ω X ′ / S - π ∗ ⁢ ω X / S , we denote by Γ 2 the integer determined by the self-intersection of Γ on X ′ .

For such normal X one defines for x ∈ X the following invariant of the 𝒪 X , x , independent of the choice of resolution:

Then a correct statement of [1, Theorem 1.4] should be: