Delta invariant of curves on rational surfaces I. The analytic approach

Abstract

We prove that if (C, 0) is a reduced curve germ on a rational surface singularity (X, 0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair C ⊂ X. Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local correction term at singular points of the global Riemann-Roch formula, valid for projective normal surfaces, introduced by Blache.

Local invariants of minimal generic curves on rational surfaces

“…Note that κ X (ℓ ′ ) in this way is defined via the analytic filtration ℓ ′ → F (ℓ ′ ), or equivalently, in terms of the Hilbert series H(t), which can be identified with the counting function of the analytic Poincaré series P (t). However, if (X, 0) is rational, then by the CDGZidentity 2.4 [8,9]. (In general, this expression is called the 'topological' κ-invariant.…”

“…The proof is based on a surgery formula of the multivariable topological Poincaré series and several recent developments in the area (see eg. [19,20,17,18,8,9]). Surprisingly, it turns out that the minimality condition imposed by being 'minimal generic' is reflected as an extremal behavior in the value of the delta invariant as well.…”

“…Then we continue with a material related with multivariable series, the topological and analytical Poincaré series, counting functions, and finally the surgery formula. We refer to [19,20,17,18,8,9] for more details. Here we also discuss in short the connection with the kappa invariant, which was historically one of the major motivations, cf.…”

“…The main goal is to recover the delta invariant δ(C) of the abstract curve (C, 0) from the topology of the embedding (C, 0) ⊂ (X, 0). We give explicit formulae whenever (C, 0) is minimal generic and (X, 0) is rational (as a continuation of [8,9]). Additionally we prove that if (X, 0) is a quotient singularity, then δ(C) only admits the values r −1 or r, where r is the number or irreducible components of (C, 0).…”

On Delta for parameterized Curve Singularities

“…The delta invariant of a reduced curve singularity has been and still is a continuous subject of research (e.g. [Hi65], [Te78], [Ca80], [Ca05], [CL06], [Ng16], [CLMN19], [IIL20], to name a few). In all these papers the curve singularities appear as fibers X s of a flat morphism X → S, and in this situation it is well known that δ(X s ) is semicontinuous, assuming that the fibers are reduced (cf.…”