On the degree of global smoothings for subanalytic sets
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909186097414144 |
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| author | Savi, Enrico |
| author_facet | Savi, Enrico |
| contents | In [4] Bierstone and Parusinski proved the existence of global smoothings for closed subanalytic sets, both in an embedded and a non-embedded sense. In particular, in the non-embedded desingularization procedure the authors construct smoothings of (generically) even degree, indeed it is well-known the existence of subanalytic sets which do not admit non-embedded smoothings of (generically) odd degree. In this paper we introduce a natural topological notion of nonbounding equator for subanalytic sets and we prove a criterion to determine whether a closed subanalytic set $X$ only admits global smoothings of even degree along the nonbounding equator. More in detail, we prove that if $X$ has a nonbounding equator $Y$ then every smoothing of $X$ which is a covering on a connected neighborhood $W$ of $Y$ has even degree over $W$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_06841 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the degree of global smoothings for subanalytic sets Savi, Enrico Algebraic Geometry 32B20, 14P15 (Primary), 32S45, 14E15, 14P05, 14P10 (Secondary) In [4] Bierstone and Parusinski proved the existence of global smoothings for closed subanalytic sets, both in an embedded and a non-embedded sense. In particular, in the non-embedded desingularization procedure the authors construct smoothings of (generically) even degree, indeed it is well-known the existence of subanalytic sets which do not admit non-embedded smoothings of (generically) odd degree. In this paper we introduce a natural topological notion of nonbounding equator for subanalytic sets and we prove a criterion to determine whether a closed subanalytic set $X$ only admits global smoothings of even degree along the nonbounding equator. More in detail, we prove that if $X$ has a nonbounding equator $Y$ then every smoothing of $X$ which is a covering on a connected neighborhood $W$ of $Y$ has even degree over $W$. |
| title | On the degree of global smoothings for subanalytic sets |
| topic | Algebraic Geometry 32B20, 14P15 (Primary), 32S45, 14E15, 14P05, 14P10 (Secondary) |
| url | https://arxiv.org/abs/2210.06841 |