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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.29157 |
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| _version_ | 1866917372314517504 |
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| author | Diamond, Benjamin E. |
| author_facet | Diamond, Benjamin E. |
| contents | To each ternary sextic $f(X_0, X_1, X_2)$ whose associated plane curve is smooth, the Shioda construction attaches a smooth sextic fourfold $X \subset \mathbb{P}^5$ whose defining equation $f(X_0, X_1, X_2) - f(Y_0, Y_1, Y_2)$ is fixed under the involution $ι: (X_0, X_1, X_2, Y_0, Y_1, Y_2) \mapsto i \cdot (Y_0, Y_1, Y_2, -X_0, -X_1, -X_2)$. The induced action $ι^* : H^4(X, \mathbb{Q}) \to H^4(X, \mathbb{Q})$ fixes a Hodge substructure $H \subset H^4(X, \mathbb{Q})$ whose Hodge coniveau is 1. By the general Hodge conjecture, we expect that there should exist a divisor $Y \subset X$ for which $H \subset \ker\left( H^4(X, \mathbb{Q}) \to H^4(X \setminus Y, \mathbb{Q}) \right)$. We verify this prediction in case the Waring rank of $f(X_0, X_1, X_2)$ takes on its minimum possible value, partially answering a question of Voisin (J. Math. Sci. Univ. Tokyo '15). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29157 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hodge Structures in Sextic Fourfolds Equipped with an Involution Diamond, Benjamin E. Algebraic Geometry To each ternary sextic $f(X_0, X_1, X_2)$ whose associated plane curve is smooth, the Shioda construction attaches a smooth sextic fourfold $X \subset \mathbb{P}^5$ whose defining equation $f(X_0, X_1, X_2) - f(Y_0, Y_1, Y_2)$ is fixed under the involution $ι: (X_0, X_1, X_2, Y_0, Y_1, Y_2) \mapsto i \cdot (Y_0, Y_1, Y_2, -X_0, -X_1, -X_2)$. The induced action $ι^* : H^4(X, \mathbb{Q}) \to H^4(X, \mathbb{Q})$ fixes a Hodge substructure $H \subset H^4(X, \mathbb{Q})$ whose Hodge coniveau is 1. By the general Hodge conjecture, we expect that there should exist a divisor $Y \subset X$ for which $H \subset \ker\left( H^4(X, \mathbb{Q}) \to H^4(X \setminus Y, \mathbb{Q}) \right)$. We verify this prediction in case the Waring rank of $f(X_0, X_1, X_2)$ takes on its minimum possible value, partially answering a question of Voisin (J. Math. Sci. Univ. Tokyo '15). |
| title | Hodge Structures in Sextic Fourfolds Equipped with an Involution |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2603.29157 |