Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.02059 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917767450460160 |
|---|---|
| author | Scully, Stephen |
| author_facet | Scully, Stephen |
| contents | Let $p$ and $q$ be anisotropic quasilinear quadratic forms over a field $F$ of characteristic $2$, and let $i$ be the isotropy index of $q$ after scalar extension to the function field of the affine quadric with equation $p=0$. In this article, we establish a strong constraint on $i$ in terms of the dimension of $q$ and two stable birational invariants of $p$, one of which is the well-known "Izhboldin dimension", and the other of which is a new invariant that we denote $Δ(p)$. Examining the contribution from the Izhboldin dimension, we obtain a result that unifies and extends the quasilinear analogues of two fundamental results on the isotropy of non-singular quadratic forms over function fields of quadrics in arbitrary characteristic due to Karpenko and Karpenko-Merkurjev, respectively. This proves in a strong way the quasilinear case of a general conjecture previously formulated by the author, suggesting that a substantial refinement of this conjecture should hold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02059 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extended Karpenko and Karpenko-Merkurjev theorems for quasilinear quadratic forms Scully, Stephen Rings and Algebras Algebraic Geometry 11E04, 14E05 Let $p$ and $q$ be anisotropic quasilinear quadratic forms over a field $F$ of characteristic $2$, and let $i$ be the isotropy index of $q$ after scalar extension to the function field of the affine quadric with equation $p=0$. In this article, we establish a strong constraint on $i$ in terms of the dimension of $q$ and two stable birational invariants of $p$, one of which is the well-known "Izhboldin dimension", and the other of which is a new invariant that we denote $Δ(p)$. Examining the contribution from the Izhboldin dimension, we obtain a result that unifies and extends the quasilinear analogues of two fundamental results on the isotropy of non-singular quadratic forms over function fields of quadrics in arbitrary characteristic due to Karpenko and Karpenko-Merkurjev, respectively. This proves in a strong way the quasilinear case of a general conjecture previously formulated by the author, suggesting that a substantial refinement of this conjecture should hold. |
| title | Extended Karpenko and Karpenko-Merkurjev theorems for quasilinear quadratic forms |
| topic | Rings and Algebras Algebraic Geometry 11E04, 14E05 |
| url | https://arxiv.org/abs/2409.02059 |