Specializations of symplectic and van Geemen--Sarti involutions on K3 surfaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914359038443520 |
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| author | Garbagnati, Alice |
| author_facet | Garbagnati, Alice |
| contents | Given a symplectic involution $ι$ on a K3 surface $X$, the desingularization $Y$ of $X/ι$ is still a K3 surface, which in general has a different Néron--Severi group. Nevertheless, if the involution is induced by the translation by a 2-torsion section on an elliptic fibration (i.e. it is a van Geemen--Sarti involution) and the Picard number is minimal, the Néron--Severi groups of $X$ and $Y$ are known to be isometric. We first determine infinitely many codimension 2 subfamilies of projective K3 surfaces with a symplectic involution (not of van Geemen--Sarti type) whose generic members satisfy $NS(X)\simeq NS(Y)$. Then, we describe the cohomological action of a van Geemen--Sarti involution and we characterize specializations of K3 surfaces with a van Geemen--Sarti involution for which it is still true that $NS(X)\simeq NS(Y)$. There is a 5-dimensional family of K3 surfaces with van Geemen--Sarti involution for which $X\simeq Y$. The K3 surfaces in such a family admit complex multiplication, and we describe its cohomological action. We briefly discuss similar problems for order 3 symplectic automorphisms induced by a translation by a 3-torsion section on an elliptic fibration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00367 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Specializations of symplectic and van Geemen--Sarti involutions on K3 surfaces Garbagnati, Alice Algebraic Geometry 14J28, 14J50 Given a symplectic involution $ι$ on a K3 surface $X$, the desingularization $Y$ of $X/ι$ is still a K3 surface, which in general has a different Néron--Severi group. Nevertheless, if the involution is induced by the translation by a 2-torsion section on an elliptic fibration (i.e. it is a van Geemen--Sarti involution) and the Picard number is minimal, the Néron--Severi groups of $X$ and $Y$ are known to be isometric. We first determine infinitely many codimension 2 subfamilies of projective K3 surfaces with a symplectic involution (not of van Geemen--Sarti type) whose generic members satisfy $NS(X)\simeq NS(Y)$. Then, we describe the cohomological action of a van Geemen--Sarti involution and we characterize specializations of K3 surfaces with a van Geemen--Sarti involution for which it is still true that $NS(X)\simeq NS(Y)$. There is a 5-dimensional family of K3 surfaces with van Geemen--Sarti involution for which $X\simeq Y$. The K3 surfaces in such a family admit complex multiplication, and we describe its cohomological action. We briefly discuss similar problems for order 3 symplectic automorphisms induced by a translation by a 3-torsion section on an elliptic fibration. |
| title | Specializations of symplectic and van Geemen--Sarti involutions on K3 surfaces |
| topic | Algebraic Geometry 14J28, 14J50 |
| url | https://arxiv.org/abs/2603.00367 |