# Pullback of tautological bundle on Veronese curve

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.

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 $$\Bbb RP^n$$.
• Could you clarify what is $d$th power of a bundle? – Lada Dudnikova Apr 5 at 18:33
• 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. – Ted Shifrin Apr 5 at 18:36
• 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? – Lada Dudnikova Apr 5 at 19:04
• LOL ... Time to do more reading. The tautological line bundle is the bundle whose fiber over $[x]\in\Bbb RP^n$ is the line $\{tx: t\in\Bbb R\}$. – Ted Shifrin Apr 5 at 19:33