1
$\begingroup$

I asked yesterday on math.stackexchange.com that if the fiber product of two vector bundles seen in general as the fiber product of two families of special algebraic varieties over a scheme $ S $ corresponds to the direct sum of the vector bundles, which object in general related to families over a scheme $ S $ corresponds to the tensor product of the vector bundles seen as two families of algebraic varieties ? ( See here for more information : https://math.stackexchange.com/questions/2777287/is-the-concept-of-vector-bundle-is-a-special-case-of-the-notion-of-family-of-alg )

@Mohan in comments, told me that he does not know such an object, and nobody else knows the answer, so I thought that it is usually the big speakers of mathoverflow.net who have the answer to this kind of Questions. Can you please tell me which object in general related to families over a scheme $ S $ corresponds to the tensor product of the vector bundles seen as two families of algebraic varieties ?

thank you in advance for your help

Improve this question
$\endgroup$
1
  • $\begingroup$ Suppose $L = \pi^*(\mathcal{O}(1))$ for some $\pi: X \to \mathbb{P}^n$. Then $L^{\otimes d}$ corresponds to the restriction of $\mathcal{O}(d)$, of course, but it is more geometric, perhaps, to think of this as the restriction of $\mathcal{O}(1)$ from the projective space that is the codomain of the $d$-uple embedding $\pi: X \to \mathbb{P}^n \to \mathbb{P}^N$. To generalize to vector bundles, you could consider maps into Grassmannians as well, for which there are $d$-uple embeddings (I know Harris goes through this in the same book you cite in your other post). $\endgroup$
    – Eric Canton
    Commented May 14, 2018 at 16:24

0

Reset to default

You must log in to answer this question.