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Let's work in $\mathbb{RP}^n$. Let $N$ be $\frac{n!}{d!(n-d)!}$

We define the Veronese map $\mu_d : \mathbb{RP}^n \to \mathbb{RP}^N$ This map

$[x_0:x_1:x_{n-1}] \mapsto [x_0..x_d, x_0..x_{d-1}x_{d+1}, x_0^d,\dotsc]$

https://en.wikipedia.org/wiki/Veronese_surface

Here the coordinates go to values of all possible monomials of degree $d$. Let $E$ be the tautological bundle on $\mathbb{RP}^N$

How can one describe $\mu_d^* E$? I know what are the gluing cocycles of $E$. $g_{ij} = {f_j \over f_i}$, where $f_i, f_j$ are the coordinates.

My thought: the map is given explicitly, so the cocycles of the pullback may be given explicitly. What is the rank of this pullback bundle?

Here is kinda similar question, but I am not quite used to sheaf terminology.

Could you give me a hint?

Let's take an example with $n=2 , d = 2$

Then the inverse map at, say $y \neq 0 $ is $[\frac{xy}{y^2}: 1: \frac{zy}{y^2}]$

The definition of a pullback of a bundle induced by morphism $f: B' \to B$ is $\{ (b',e) : f(b') = \pi(e)\}$

If you had $(x^2:1:z^2:x:z:xz)$ and get to $(1:y^2:z^2:y:yz:z)$ then you have $g_{ij} = \frac{y^2}{x^2}$ Since they are the coordinates itself on a Veronese curve, that's a tautological bundle there, but on the $P^n$ that's $2nd$ tensor product of tautological bundle.

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The pullback of a line bundle will of course be a line bundle. And you will get the $d$th power of the tautological bundle on $\Bbb RP^n$. It's probably easier to think about the hyperplane bundle (the dual of the tautological bundle), whose sections are homogeneous polynomials of degree $1$. Each of those pulls back to a homogeneous polynomial of degree $d$ on $\Bbb RP^n$.

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  • $\begingroup$ Could you clarify what is $d$th power of a bundle? $\endgroup$ – Lada Dudnikova Apr 5 at 18:33
  • $\begingroup$ For line bundles, the transition functions of the tensor product of two bundles is the product of the transition functions. So here you take the $d$th power of the transition functions. $\endgroup$ – Ted Shifrin Apr 5 at 18:36
  • $\begingroup$ I thought, that Tautological bundle of $\mathbb{RP}^n$ is of rank $n$ and we take one coordinate say, $x_0 = 1$ and others as $({\frac{x_1}{ x_0}, \frac{ x_2}{x_0}, \ etc)}$. Then, what is a tautological bundle? $\endgroup$ – Lada Dudnikova Apr 5 at 19:04
  • $\begingroup$ LOL ... Time to do more reading. The tautological line bundle is the bundle whose fiber over $[x]\in\Bbb RP^n$ is the line $\{tx: t\in\Bbb R\}$. $\endgroup$ – Ted Shifrin Apr 5 at 19:33
  • $\begingroup$ I finally understand, will write in the body of a question. $\endgroup$ – Lada Dudnikova Apr 6 at 9:39

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