Let S be a noetherian scheme and f : X → S be a smooth morphism of relative dimension 1. For a locally constant sheaf on the complement of a divisor in X flat over S, Deligne and Laumon proved that the universal local acyclicity is equivalent to the local constancy of Swan conductors. In this article, assuming the universal local acyclicity, we show an analogous result of the continuity of local epsilon factors. We also give a generalization of this result to a family of isolated singularities.

Let X be a smooth variety over a finite field F q . Let ℓ be a rational prime number invertible in F q . For an ℓ-adic sheaf F on X, we construct a cycle supported on the singular support of F whose coefficients are ℓ-adic numbers modulo roots of unity.It is a refinement of the characteristic cycle CC(F), in the sense that it satisfies a Milnor-type formula for local epsilon factors. After establishing fundamental results on the cycles, we prove a product formula of global epsilon factors modulo roots of unity. We also give a generalization of the results to varieties over general perfect fields.• For the ℓ-adic formalism of a noetherian topos T , we refer to [7], which we review in the appendix. The derived category of constructible complexes of E-sheaves (resp. O E -sheaves) on T is denoted by D b c (T, E) (resp. D b c (T, O E )). We put 0 for objects of D b c (T, O E ) (i.e. F 0 ∈ D b c (T, O E )) and denote F := F 0 ⊗ O E E.

We propose another proof of the geometric class field theory for curves by considering blow-ups of symmetric products of curves. IntroductionLet k be a perfect field, and C be a projective smooth geometrically connected curve over k. The geometric class field theory gives a geometric description of abelian coverings of C by using generalized jacobian varieties. Let us recall its precise statement. Fix a modulus m, i.e. an effective Cartier divisor of C and let U be its complement in C. Let Pic 0 C,m be the corresponding generalized jacobian variety. Let G 0 → Pic 0 C,m be anétale isogeny of smooth commutative algebraic groups and G 1 → Pic 1 C,m be a compatible morphism of torsors. We call such a pair (G 0 → Pic 0 C,m , G 1 → Pic 1 C,m ) a covering of (Pic 0 C,m , Pic 1 C,m ). We call such a covering connected abelian if G 0 is connected and G 0 → Pic 0 C,m is an abelian isogeny. There is a natural map from U to Pic 1 C,m sending a point of U to its associated invertible sheaf with a trivialization. The geometric class field theory states:Theorem 1.1. Let C be a projective smooth geometrically connected curve over a perfect field k. Fix a modulus m of C and denote its complement by U. Let Pic 0 C,m be the generalized jacobian variety with modulus m. Then a connected abelian covering (G 0 → Pic 0 C,m , G 1 → Pic 1 C,m ) pulls back by the natural map U → Pic 1 C,m to a geometrically connected abelian covering of U whose ramification is bounded by m. Conversely, every such covering is obtained in this way.Originally this theorem was proved by M. Rosenlicht [7]. S. Lang [5] generalized his results to an arbitrary algebraic variety. Their works are explained in detail in Serre's book [8].On the other hand, in 1980s, P. Deligne found another proof for the tamely ramified case by using symmetric powers of curves. The aim of this paper is to complete his proof by considering blow-ups of symmetric powers of curves.Recently Q. Guignard did a similar work to this paper, although we do not know his results in detail.Actually we prove a variant of Theorem 1.1 now stated.Theorem 1.2. There is an isomorphism of groups between the subgroup of H 1 (U, Q/Z) consisting of a character χ such that Sw P (χ) ≤ n P − 1 for all points P ∈ m, where n P is the multiplicity of m at P , and the subgroup of H 1 (Pic C,m , Q/Z) consisting of ρ which is multiplicative, i.e. the self-external product ρ ⊠ 1 + 1 ⊠ ρ on Pic C,m × k Pic C,m equals to m * ρ, the pull back of ρ by the multiplication map m : Pic C,m × k Pic C,m → Pic C,m .The relation between Theorem 1.1 and Theorem 1.2 will be explained in Section 4. When k is algebraically closed, Theorem 1.2 can be stated as follows. Let ρ be a multiplicative character of Pic C,m . Fix a closed point P ∈ Pic 1 C,m . The multiplicativity of ρ implies that, for an integer d, the pull back of ρ d by the multiplication by dP Pic 0 C,m → Pic d C,m coincides with ρ 0 . In this way, Theorem 1.2 can be restated as below:Theorem 1.3. Assume that k is algebraically closed. Then there is an isomorphism of grou...