The correction term for the Riemann–Roch formula of cyclic quotient singularities and associated invariants

2019

Preprint

Self Cite

In [9], Esnault-Viehweg developed the theory of cyclic branched coverings X → X of smooth surfaces providing a very explicit formula for the decomposition of H 1 ( X, C) in terms of a resolution of the ramification locus. Later, in [1] the first author applies this to the particular case of coverings of P 2 reducing the problem to a combination of global and local conditions on projective curves.In this paper we extend the above results in three directions: first, the theory is extended to surfaces with abelian quotient singularities, second the ramification locus can be partially resolved and need not be reduced, and finally global and local conditions are given to describe the irregularity of cyclic branched coverings of the weighted projective plane.The techniques required for these results are conceptually different and provide simpler proofs for the classical results. For instance, the local contribution comes from certain modules that have the flavor of quasi-adjunction and multiplier ideals on singular surfaces.As an application, a Zariski pair of curves on a singular surface is described. In particular, we prove the existence of two cuspidal curves of degree 12 in the weighted projective plane P 2(1,1,3) with the same singularities but non-homeomorphic embeddings. This is shown by proving that the cyclic covers of P 2(1,1,3) of order 12 ramified along the curves have different irregularity. In the process, only a partial resolution of singularities is required.

“…Step 4. Finally, combining properties ( 5), (6), and (7) in Lemma 4.6 together with Step 1 above, it immediately follows that β…”

Section: Global Sections and Weighted Blow-upsmentioning

confidence: 70%

“…M j := (∆w 2 + 1) deg w C j , for all j = 1, .., r, then M j satisfies (6). By construction M = (∆w 2 +1) deg w C which proves (5) and M deg w C = ∆w 2 + 1 satisfies (7).…”

Section: Global Sections and Weighted Blow-upsmentioning

confidence: 74%

Section: Theorem 115 ([4]mentioning

confidence: 99%

See 1 more Smart Citation

Cyclic branched coverings of surfaces with abelian quotient singularities

Bartolo¹,

Cogolludo-Agustín²,

2019

Preprint

Self Cite

“…It turns out that 'minimal generic' curve germs on a cyclic quotient singularity are all ordinary r-tuples, a fact firstly noticed in [11] (a work, which partly motivated our work).…”

Section: Introductionmentioning

confidence: 67%

“…and all the components of C are smooth and do not intersect each other, and they intersect E transversally. Also, a curve (C, 0) ⊂ (X, 0) is called minimal generic if it is a minimal generic h-curve for some h ∈ H. Note that minimal generic curves were introduced as generic curves in [11] and studied in the context of cyclic quotient singularities, however we think the term minimal generic is more appropriate.…”

Section: Preliminariesmentioning

confidence: 99%

Local invariants of minimal generic curves on rational surfaces

Cogolludo-Agustín¹,

Tamás²,

et al. 2020

Preprint

Self Cite

Let (C, 0) be a reduced curve germ in a normal surface singularity (X, 0). 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). (δ(C) = r −1 realizes the extremal lower bound, valid only for 'ordinary r-tuples'.

Local invariants of minimal generic curves on rational surfaces

Cogolludo-Agustín¹,

Tamás²,

𝑝-Adic Analysis, Arithmetic and Singularities

et al. 2022

Let ( C , 0 ) (C,0) be a reduced curve germ in a normal surface singularity ( X , 0 ) (X,0) . The main goal is to recover the delta invariant δ ( C ) \delta (C) of the abstract curve ( C , 0 ) (C,0) from the topology of the embedding ( C , 0 ) ⊂ ( X , 0 ) (C,0)\subset (X,0) . We give explicit formulae whenever ( C , 0 ) (C,0) is minimal generic and ( X , 0 ) (X,0) is rational (as a continuation of some previous papers by the authors). Additionally, in this case, we prove that if ( X , 0 ) (X,0) is a quotient singularity, then δ ( C ) \delta (C) only admits the values r − 1 r-1 or r r , where r r is the number or irreducible components of ( C , 0 ) (C,0) . ( δ ( C ) = r − 1 \delta (C)=r-1 realizes the extremal lower bound, valid only for ‘ordinary r r –tuples’.)