Extendability of general $K3$ surfaces without Gaussian maps and classification of non-prime Fano threefolds
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913884463431680 |
|---|---|
| author | Bangere, Purnaprajna Mukherjee, Jayan |
| author_facet | Bangere, Purnaprajna Mukherjee, Jayan |
| contents | In arXiv:2409.03960, we introduced an approach to the question of extendability of projective varieties via degeneration to ribbons. In this article we build on these methods to give a new proof of optimal results on the extendability of general non-prime $K3$ surfaces, classification of non-prime Fano threefolds and Mukai varieties and the irreducibility of their Hilbert schemes. The methods in this article also show the non-extendability of prime $K3$ surfaces for infinitely many values of $g$, for example when $g$ is of the form $g = 4k+1$, $k \geq 5$. This involves degenerations of $K3$ surfaces to ribbons on embedded Hirzebruch surfaces, called $K3$ carpets. We directly give optimal upper bounds on the cohomology of the twisted normal bundle of the $K3$ carpets instead of computing coranks of Gaussian maps of the canonical curve sections. As a result of independent interest, we show such $K3$ carpets also appear as degenerations of smoothable simple normal crossings of two Hirzebruch surfaces embedded by arbitrary linear series intersecting along an anticanonical elliptic curve. Such type II degenerations constitute a smooth locus of codimension $6$ in the Hilbert scheme of $K3$ surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12661 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extendability of general $K3$ surfaces without Gaussian maps and classification of non-prime Fano threefolds Bangere, Purnaprajna Mukherjee, Jayan Algebraic Geometry 14B10, 14N05, 14J28, 14J45 In arXiv:2409.03960, we introduced an approach to the question of extendability of projective varieties via degeneration to ribbons. In this article we build on these methods to give a new proof of optimal results on the extendability of general non-prime $K3$ surfaces, classification of non-prime Fano threefolds and Mukai varieties and the irreducibility of their Hilbert schemes. The methods in this article also show the non-extendability of prime $K3$ surfaces for infinitely many values of $g$, for example when $g$ is of the form $g = 4k+1$, $k \geq 5$. This involves degenerations of $K3$ surfaces to ribbons on embedded Hirzebruch surfaces, called $K3$ carpets. We directly give optimal upper bounds on the cohomology of the twisted normal bundle of the $K3$ carpets instead of computing coranks of Gaussian maps of the canonical curve sections. As a result of independent interest, we show such $K3$ carpets also appear as degenerations of smoothable simple normal crossings of two Hirzebruch surfaces embedded by arbitrary linear series intersecting along an anticanonical elliptic curve. Such type II degenerations constitute a smooth locus of codimension $6$ in the Hilbert scheme of $K3$ surfaces. |
| title | Extendability of general $K3$ surfaces without Gaussian maps and classification of non-prime Fano threefolds |
| topic | Algebraic Geometry 14B10, 14N05, 14J28, 14J45 |
| url | https://arxiv.org/abs/2503.12661 |