0
$\begingroup$

I'm doing the exercises of the book Fulton Algebraic curves and I'm stucked in the following problem:

A subset $V\subset\mathbb{P}^n(k)$ is a linear subvariety of $\mathbb{P}^n(k)$ if $V=V(H_1,\dots,H_r)$, where each $H_i$ is a form of degree $1$.

a) Show that if $T$ is a projective change of coordinates, then $V^T=T^{-1}(V)$ is also a linear subvariety.

b) Show that exists a projective change of coordinates $T$ of $\mathbb{P}^n$ such that $V^T=V(x_{m+2},\dots,x_{n+1})$, then $V$ is a variety.

c) Show that the $m$ that appears in part (b) is independent of the choice of $T$.

Can anyone give me a Hint to do the part (b) and (c)?

Any help will be appreciated!

Thanks in advance !

$\endgroup$
0
$\begingroup$

Hint: Linear Algebra. Namely, hyperplanes are the zero locus of linear forms that can be regarded as elements of the dual vector space $(k^{n+1})^*$. Keep in mind the theorem in linear algebra that claims that given a set $\lambda_1, \cdots, \lambda_r$ of linearly independent $1$-forms then there exists another $n+1-r$ $1$-forms $\lambda_{r+1},\cdots,\lambda_{n+1}$ such that all together $\lambda_1,\cdots,\lambda_{n+1}$ are a basis of $(k^{n+1})^*$. If the hint is not sufficient for you I can add more details.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.