Gorenstein Simplices and Even Binary Self-Complementary Codes
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910171933966336 |
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| author | Tsuchiya, Akiyoshi |
| author_facet | Tsuchiya, Akiyoshi |
| contents | It is known that if a Gorenstein simplex of dimension \(d\) and degree \(s\) is not a lattice pyramid, then \(d \leq 2s-1\). In this paper, we study the extremal case \(d=2s-1\). More precisely, we characterize Gorenstein simplices of dimension \(2s-1\) and degree \(s\) which are not lattice pyramids in terms of even binary self-complementary codes. As an application, combining this characterization with existing classification results on reflexive simplices, we classify Gorenstein simplices of degree \(3\) and \(4\). Equivalently, we classify polarized \(d\)-dimensional Gorenstein fake weighted projective spaces \((X,L)\) satisfying $-K_X=(d-2)L$ or $-K_X=(d-3)L$, where \(-K_X\) is the anticanonical divisor of \(X\) and \(L\) is a Cartier divisor on \(X\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15005 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gorenstein Simplices and Even Binary Self-Complementary Codes Tsuchiya, Akiyoshi Combinatorics Algebraic Geometry 52B20 (Primary), 14J45, 14M25, 94B05 (Secondary) It is known that if a Gorenstein simplex of dimension \(d\) and degree \(s\) is not a lattice pyramid, then \(d \leq 2s-1\). In this paper, we study the extremal case \(d=2s-1\). More precisely, we characterize Gorenstein simplices of dimension \(2s-1\) and degree \(s\) which are not lattice pyramids in terms of even binary self-complementary codes. As an application, combining this characterization with existing classification results on reflexive simplices, we classify Gorenstein simplices of degree \(3\) and \(4\). Equivalently, we classify polarized \(d\)-dimensional Gorenstein fake weighted projective spaces \((X,L)\) satisfying $-K_X=(d-2)L$ or $-K_X=(d-3)L$, where \(-K_X\) is the anticanonical divisor of \(X\) and \(L\) is a Cartier divisor on \(X\). |
| title | Gorenstein Simplices and Even Binary Self-Complementary Codes |
| topic | Combinatorics Algebraic Geometry 52B20 (Primary), 14J45, 14M25, 94B05 (Secondary) |
| url | https://arxiv.org/abs/2604.15005 |