Maps between projective spaces induced by singular matrices

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In this question, $V(a,k)$ denotes the $a$-dimensional vector space over the field $k$. Now consider $V(b,k)$ and $V(c,k)$, and let $M$ be a singular $(c times b)$-matrix over $k$. Let $ell_M: V(b,k) mapsto V(c,k)$ be the linear transformation induced by $M$.
What kind of map does $ell_M$ induce between the projective spaces $mathbbP^b - 1(k)$ and $mathbbP^c - 1(k)$ (seen as projective varieties)? I guess it is a rational map, but what are the most interesting, or useful/important, examples of maps of this type ?




algebraic-geometry projective-space projective-schemes affine-varieties birational-geometry




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    This is a projection of $mathbf P^b-1$ onto some $mathbf P^d$ followed by a linear embedding of $mathbf P^d$ into $mathbf P^c-1$. The centre of the projection is the linear subspace of $mathbf P^b-1$ obtained by projectivising the kernel of $M$.
    – Asal Beag Dubh
    Aug 23 at 11:08










  • @AsalBeagDubh : Could you put this in the form of an answer, with more details, if possible ? Thanks !
    – Boccherini
    Aug 23 at 11:53














up vote
0
down vote

favorite












In this question, $V(a,k)$ denotes the $a$-dimensional vector space over the field $k$. Now consider $V(b,k)$ and $V(c,k)$, and let $M$ be a singular $(c times b)$-matrix over $k$. Let $ell_M: V(b,k) mapsto V(c,k)$ be the linear transformation induced by $M$.
What kind of map does $ell_M$ induce between the projective spaces $mathbbP^b - 1(k)$ and $mathbbP^c - 1(k)$ (seen as projective varieties)? I guess it is a rational map, but what are the most interesting, or useful/important, examples of maps of this type ?




algebraic-geometry projective-space projective-schemes affine-varieties birational-geometry




share|cite|improve this question
















  • 1




    This is a projection of $mathbf P^b-1$ onto some $mathbf P^d$ followed by a linear embedding of $mathbf P^d$ into $mathbf P^c-1$. The centre of the projection is the linear subspace of $mathbf P^b-1$ obtained by projectivising the kernel of $M$.
    – Asal Beag Dubh
    Aug 23 at 11:08










  • @AsalBeagDubh : Could you put this in the form of an answer, with more details, if possible ? Thanks !
    – Boccherini
    Aug 23 at 11:53












up vote
0
down vote

favorite









up vote
0
down vote

favorite











In this question, $V(a,k)$ denotes the $a$-dimensional vector space over the field $k$. Now consider $V(b,k)$ and $V(c,k)$, and let $M$ be a singular $(c times b)$-matrix over $k$. Let $ell_M: V(b,k) mapsto V(c,k)$ be the linear transformation induced by $M$.
What kind of map does $ell_M$ induce between the projective spaces $mathbbP^b - 1(k)$ and $mathbbP^c - 1(k)$ (seen as projective varieties)? I guess it is a rational map, but what are the most interesting, or useful/important, examples of maps of this type ?




algebraic-geometry projective-space projective-schemes affine-varieties birational-geometry




share|cite|improve this question












In this question, $V(a,k)$ denotes the $a$-dimensional vector space over the field $k$. Now consider $V(b,k)$ and $V(c,k)$, and let $M$ be a singular $(c times b)$-matrix over $k$. Let $ell_M: V(b,k) mapsto V(c,k)$ be the linear transformation induced by $M$.
What kind of map does $ell_M$ induce between the projective spaces $mathbbP^b - 1(k)$ and $mathbbP^c - 1(k)$ (seen as projective varieties)? I guess it is a rational map, but what are the most interesting, or useful/important, examples of maps of this type ?





algebraic-geometry projective-space projective-schemes affine-varieties birational-geometry





share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 23 at 10:05









Boccherini

1779




1779







  • 1




    This is a projection of $mathbf P^b-1$ onto some $mathbf P^d$ followed by a linear embedding of $mathbf P^d$ into $mathbf P^c-1$. The centre of the projection is the linear subspace of $mathbf P^b-1$ obtained by projectivising the kernel of $M$.
    – Asal Beag Dubh
    Aug 23 at 11:08










  • @AsalBeagDubh : Could you put this in the form of an answer, with more details, if possible ? Thanks !
    – Boccherini
    Aug 23 at 11:53












  • 1




    This is a projection of $mathbf P^b-1$ onto some $mathbf P^d$ followed by a linear embedding of $mathbf P^d$ into $mathbf P^c-1$. The centre of the projection is the linear subspace of $mathbf P^b-1$ obtained by projectivising the kernel of $M$.
    – Asal Beag Dubh
    Aug 23 at 11:08










  • @AsalBeagDubh : Could you put this in the form of an answer, with more details, if possible ? Thanks !
    – Boccherini
    Aug 23 at 11:53







1




1




This is a projection of $mathbf P^b-1$ onto some $mathbf P^d$ followed by a linear embedding of $mathbf P^d$ into $mathbf P^c-1$. The centre of the projection is the linear subspace of $mathbf P^b-1$ obtained by projectivising the kernel of $M$.
– Asal Beag Dubh
Aug 23 at 11:08




This is a projection of $mathbf P^b-1$ onto some $mathbf P^d$ followed by a linear embedding of $mathbf P^d$ into $mathbf P^c-1$. The centre of the projection is the linear subspace of $mathbf P^b-1$ obtained by projectivising the kernel of $M$.
– Asal Beag Dubh
Aug 23 at 11:08












@AsalBeagDubh : Could you put this in the form of an answer, with more details, if possible ? Thanks !
– Boccherini
Aug 23 at 11:53




@AsalBeagDubh : Could you put this in the form of an answer, with more details, if possible ? Thanks !
– Boccherini
Aug 23 at 11:53















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