How to visualize the $d$-uple embedding?

Question

Given positive integers $n$ and $d$, let $\{M_i\}_{i \in \{0 \dotsc N\}}$ be the collection of all $N+1$ monomials in $n+1$ variables of degree $d$. Then given a point $a = [a_0 : \dotsb : a_n]$ we can define the map \begin{align*} \boldsymbol{P}^{n} &\to \boldsymbol{P}^{N} \\ [a_0 : \dotsb : a_n] &\mapsto [M_1(a) : \dotsb : M_N(a)] \,. \end{align*} This map is called the $d$-uple embedding of $\boldsymbol{P}^{n}$ into $\boldsymbol{P}^{N}$.

Is there any good way to visualize what this $d$-uple embedding looks like? What is the motivation for this construction?

Answer 1

I guess that by "d-Uple embedding" you mean the Veronese embedding of $\mathbb{P}_{\mathbb{C}}^{1}$ in $\mathbb{P}_{\mathbb{C}}^{d}$. If this is the case, taking $d=3$ you have that the image of the embedding is the twisted cubic curve $V(XZ-Y^{2},YW-Z^{2},XW-YZ)\subseteq\mathbb{P}_{\mathbb{C}}^{3}$, that in the affine chart $\{W=1\}$ has the following parametric form: $\{(t,t^{2},t^{3})| t\in\mathbb{C}\}$; looking only at its real points, you can easily draw or plot it.

If you are interested in Veronese embeddings of projective spaces of greater dimension, a quite famous example is the 2-uple embedding of $\mathbb{P}_{\mathbb{C}}^{2}$ in $\mathbb{P}_{\mathbb{C}}^{5}$, that is called the Veronese surface. You can find plenty of images of it!

Answer 2

A motivation for the Veronese embedding is the following : we understand more or less hyperplane section of variety. So if we can express a variety as hyperplane section of another variety, this can give us more information.

Now, the hypersurface $\sum_i a_i m_i = 0$ where $m_i$ are all monomials can be seen as an hyperplane $\sum_i a_i X_i = 0$ of $P^N$ intersected with the Veronese variety $X = v_d(P^n)$ ! This trick has lot of applications, for example the Veronese embedding + the Lefschetz hyperplane theorem gives you that any smooth hypersurface has the same Betti numbers as $P^{n-1}$, so $b_i = 1$ if $i$ is even and $b_i = 0$ for $i$ odd or $i > n$ , except for $i = \dim X$ (i.e the middle Betti numbers).