I'm working through Vakil and Beauville's notes on Rational Surfaces; there are a few details which are eluding me for the time being. I think the foundations are that we want to consider $\mathbb{P}^{2}$, with $H$ denoting the divisor class of a line in $\mathbb{P}^{2}$. Let's now blow-up $\mathbb{P}^{2}$ at $n$ distinct points $p_{1}, \ldots, p_{n}$ with exceptional divisors $E_{1}, \ldots, E_{n}$. Vakil states that the sections of the line bundle $\mathcal{O}(aH-E_{1} - \ldots - E_{n})$ correspond to a linear system of degree $a$ divisors of $\mathbb{P}^{2}$ which contain the points $p_{1}, \ldots, p_{n}$. This much I think I understand. And even more simply, the full linear system of conics on $\mathbb{P}^{2}$ is simply $\mathcal{O}(2H)$ which of course corresponds to the Veronese embedding $\mathbb{P}^{2} \hookrightarrow \mathbb{P}^{5}$. In the intersection ring, we have the relations $H^{2}=1$, $E_{i} \cdot E_{j}=0$, $H \cdot E_{i}=0$, and $E_{i}^{2}=0$.
My first confusion is the following. Of course both linear systems given above provide embeddings into larger projective spaces. The degree of this embedding is apparently given, in the case of $\mathbb{P}^{2} \hookrightarrow \mathbb{P}^{5}$ as $2H \cdot 2H=4$. I see clearly where the 4 comes from, but why did we square the class $2H$? This is indeed consistent with the Veronese surface being degree 4.
Moreover, if you consider all conics in $\mathbb{P}^{2}$ passing through some fixed point $p_{1}$, you consider divisors $2H-E_{1}$, and square them to get a degree 3 surface in $\mathbb{P}^{4}$. Again, I am not seeing why we square the class.
Finally, I think this cubic surface in $\mathbb{P}^{4}$ arises via projection away from a point in the Veronese surface. Can someone help me with some geometric intuition of this, and how it may relate to the above? I'm guessing the point in the Veronese surface from which we project away, should be the image of the point in $\mathbb{P}^{2}$ through which all conics pass.