Skip to main content
MathOverflow

Return to Question

deleted 9 characters in body
Source Link
nxir
nxir
  • 1.5k
  • 8
  • 14

Let $A$ be a central, simple algebra of degree $2n$ with orthogonal involution $\sigma$.

Let SB$(A)$ the Severi-Brauer variety, which is the variety of left ideals of reduced dimension one in $A$.

The involution variety Inv$(A,\sigma)= \{ I \in $SB$(A)$ $|$ $\sigma(I)I = 0\}$ is known to be a hyperplane section of SB$(A)$.

We define a generalized SB$(A)_{l,n}$ as the variety of left ideals in $A$ of reduced dimension $l$, when $deg(A) = n$. In an analogue way one can define a generalized involution variety Inv$(A,\sigma)_{l,n}$.

I asked myself, if for $l$ other than $1$, Inv$(A,\sigma)_{l,n}$ is a hyperplane section of SB$(A)_{l,n}$. However calculatiingCalculating the dimensions i think i found that this is not possible. However we still can ask

Question: For central, simple algebras $A,B$ of degrees $n,2m$, and for $\sigma$ beeing an orthogonal involution on $B$. Can there be $l,k$ such that Inv$(B,\sigma)_{k,2m}$ is a hyperplane section of SB$(A)_{l,n}$ ?

What inspired me to think about such things is a result from

D. Krashen - Zero cycles on homogeneous varieties,

stating that for $A$ of degree $4$ and $B$ of degree $6$ there is an isomorphism between Inv$(B,\sigma)_{1,6}$ and SB$(A)_{2,4}$.

Let $A$ be a central, simple algebra of degree $2n$ with orthogonal involution $\sigma$.

Let SB$(A)$ the Severi-Brauer variety, which is the variety of left ideals of reduced dimension one in $A$.

The involution variety Inv$(A,\sigma)= \{ I \in $SB$(A)$ $|$ $\sigma(I)I = 0\}$ is known to be a hyperplane section of SB$(A)$.

We define a generalized SB$(A)_{l,n}$ as the variety of left ideals in $A$ of reduced dimension $l$, when $deg(A) = n$. In an analogue way one can define a generalized involution variety Inv$(A,\sigma)_{l,n}$.

I asked myself, if for $l$ other than $1$, Inv$(A,\sigma)_{l,n}$ is a hyperplane section of SB$(A)_{l,n}$. However calculatiing the dimensions i think i found that this is not possible. However we still can ask

Question: For central, simple algebras $A,B$ of degrees $n,2m$, and for $\sigma$ beeing an orthogonal involution on $B$. Can there be $l,k$ such that Inv$(B,\sigma)_{k,2m}$ is a hyperplane section of SB$(A)_{l,n}$ ?

What inspired me to think about such things is a result from

D. Krashen - Zero cycles on homogeneous varieties,

stating that for $A$ of degree $4$ and $B$ of degree $6$ there is an isomorphism between Inv$(B,\sigma)_{1,6}$ and SB$(A)_{2,4}$.

Let $A$ be a central, simple algebra of degree $2n$ with orthogonal involution $\sigma$.

Let SB$(A)$ the Severi-Brauer variety, which is the variety of left ideals of reduced dimension one in $A$.

The involution variety Inv$(A,\sigma)= \{ I \in $SB$(A)$ $|$ $\sigma(I)I = 0\}$ is known to be a hyperplane section of SB$(A)$.

We define a generalized SB$(A)_{l,n}$ as the variety of left ideals in $A$ of reduced dimension $l$, when $deg(A) = n$. In an analogue way one can define a generalized involution variety Inv$(A,\sigma)_{l,n}$.

I asked myself, if for $l$ other than $1$, Inv$(A,\sigma)_{l,n}$ is a hyperplane section of SB$(A)_{l,n}$. Calculating the dimensions i think i found that this is not possible. However we still can ask

Question: For central, simple algebras $A,B$ of degrees $n,2m$, and for $\sigma$ beeing an orthogonal involution on $B$. Can there be $l,k$ such that Inv$(B,\sigma)_{k,2m}$ is a hyperplane section of SB$(A)_{l,n}$ ?

What inspired me to think about such things is a result from

D. Krashen - Zero cycles on homogeneous varieties,

stating that for $A$ of degree $4$ and $B$ of degree $6$ there is an isomorphism between Inv$(B,\sigma)_{1,6}$ and SB$(A)_{2,4}$.

Source Link
nxir
nxir
  • 1.5k
  • 8
  • 14

Are Generalized Involution Varieties hyperlane sections of Generalized Severi Brauer Varieties?

Let $A$ be a central, simple algebra of degree $2n$ with orthogonal involution $\sigma$.

Let SB$(A)$ the Severi-Brauer variety, which is the variety of left ideals of reduced dimension one in $A$.

The involution variety Inv$(A,\sigma)= \{ I \in $SB$(A)$ $|$ $\sigma(I)I = 0\}$ is known to be a hyperplane section of SB$(A)$.

We define a generalized SB$(A)_{l,n}$ as the variety of left ideals in $A$ of reduced dimension $l$, when $deg(A) = n$. In an analogue way one can define a generalized involution variety Inv$(A,\sigma)_{l,n}$.

I asked myself, if for $l$ other than $1$, Inv$(A,\sigma)_{l,n}$ is a hyperplane section of SB$(A)_{l,n}$. However calculatiing the dimensions i think i found that this is not possible. However we still can ask

Question: For central, simple algebras $A,B$ of degrees $n,2m$, and for $\sigma$ beeing an orthogonal involution on $B$. Can there be $l,k$ such that Inv$(B,\sigma)_{k,2m}$ is a hyperplane section of SB$(A)_{l,n}$ ?

What inspired me to think about such things is a result from

D. Krashen - Zero cycles on homogeneous varieties,

stating that for $A$ of degree $4$ and $B$ of degree $6$ there is an isomorphism between Inv$(B,\sigma)_{1,6}$ and SB$(A)_{2,4}$.