Obrifold del Pezzo surfaces in $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$ format
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866915357780869120 |
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| author | Qureshi, Muhammad Imran |
| author_facet | Qureshi, Muhammad Imran |
| contents | We construct two types of wellformed and quasismooth biregular models (infinite series) of rigid orbifold del Pezzo surfaces having their (sub) anti-canonical embeddings in $\mathbb P^6(w_i) $. One type of model contains a family of rigid del Pezzo surfaces with a fixed Fano index and weights of ambient $\mathbb P^6(w_i)$ are parameterized by positive integers. In the other type of models, weights of $\mathbb P^6(w_i)$ and Fano index, both are parameterized by the positive integers. The equations describing their images under (sub) anti-canonical embeddings are given in terms of the equations of the Segre embedding of $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$, which has codimension 4 in $\mathbb P^7$. We also give a formula for the Hilbert series of a generic weighted $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$ variety, a key tool in these constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17339 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Obrifold del Pezzo surfaces in $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$ format Qureshi, Muhammad Imran Algebraic Geometry 14J45, 14Q10 We construct two types of wellformed and quasismooth biregular models (infinite series) of rigid orbifold del Pezzo surfaces having their (sub) anti-canonical embeddings in $\mathbb P^6(w_i) $. One type of model contains a family of rigid del Pezzo surfaces with a fixed Fano index and weights of ambient $\mathbb P^6(w_i)$ are parameterized by positive integers. In the other type of models, weights of $\mathbb P^6(w_i)$ and Fano index, both are parameterized by the positive integers. The equations describing their images under (sub) anti-canonical embeddings are given in terms of the equations of the Segre embedding of $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$, which has codimension 4 in $\mathbb P^7$. We also give a formula for the Hilbert series of a generic weighted $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$ variety, a key tool in these constructions. |
| title | Obrifold del Pezzo surfaces in $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$ format |
| topic | Algebraic Geometry 14J45, 14Q10 |
| url | https://arxiv.org/abs/2312.17339 |