Enumeração de superfícies em P³ singulares ao longo de curvas redutíveis
This thesis investigates families of singular surfaces in \(\mathbb{P}^3\) along certain reducible curves. Let \(\mathbb{W}_1,\mathbb{W}_2,\dots ,\mathbb{W}_l\) denote closed, irreducible subvarieties of a Hilbert scheme, \(Hilb_{P_{\mathbb{W}_i}(t)}(\mathbb{P}^3)\), of curves in \(\mathbb{P}^3\) wi...
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| Format: | doctoral thesis |
| Status: | Published version |
| Publication Date: | 2023 |
| Country: | Brasil |
| Institution: | Universidade Federal de Minas Gerais (UFMG) |
| Repository: | Repositório Institucional da UFMG |
| Language: | Portuguese |
| OAI Identifier: | oai:repositorio.ufmg.br:1843/81192 |
| Online Access: | http://hdl.handle.net/1843/81192 |
| Access Level: | Open access |
| Keyword: | Geometria enumerativa Fórmula de resíduos de Bott Singularidades Matemática - Teses Geometria algébrica - Teses Geometria enumerativa – Teses Singularidades (Matemática) – Teses |
| Summary: | This thesis investigates families of singular surfaces in \(\mathbb{P}^3\) along certain reducible curves. Let \(\mathbb{W}_1,\mathbb{W}_2,\dots ,\mathbb{W}_l\) denote closed, irreducible subvarieties of a Hilbert scheme, \(Hilb_{P_{\mathbb{W}_i}(t)}(\mathbb{P}^3)\), of curves in \(\mathbb{P}^3\) with Hilbert polynomial \(P_{\mathbb{W}_i}(t),i=1,\dots,l\). Let \(\mathbb{W}'\) be the closure in \(Hilb_{P_{\mathbb{W}'}(t)}(\mathbb{P}^3)\) of the family of subschemes of \(\mathbb{P}^3\) defined by ideals of the form \(\mathcal{I}_{\underline{W}}:=(\mathcal{I}_{W_1})^2\cap\dots\cap(\mathcal{I}_{W_l})^2\), where \(W_i\) is a generic member of \(\mathbb{W}_i\), for all \(i=1,\dots ,l\). We show that for \(d\gg0\) the general member of \(\mathbb{P}(H^0(\mathcal{I}_{\underline{W}}(d)))\) is a surface of degree \(d\) in \(\mathbb{P}^{3}\) singular along \(W':=W_1\cup \dots \cup W_l\). We consider the subvariety \(\Sigma(\mathbb{W}', d) \subset \mathbb{P}^{N_d} = \mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^{3}}(d)))\), formed by surfaces of degree \(d\) in \(\mathbb{P}^{3}\), which are singular along some \(W'\) as above. We show that the degree of \(\Sigma(\mathbb{W}', d)\) is given by a polynomial \(p^{\mathbb{W}'}(d)\) for \(d\gg0\). We are able to compute explicitly \(p^{\mathbb{W}'}(d)\) for certain \(\mathbb{W}'\). We state a conjecture about the degree of \(p^{\mathbb{W}'}(d)\), verified for the families worked out here. |
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