Good Morning,

I've been trying to brush up a bit on linear systems lately, and I've ran into the following (seemingly) contradictory statements. Hopefully someone can tell me where I'm going wrong here (and say how to go right).

Consider $H^0(\mathbb{P}^1, \mathcal{O}(3))$ which is the $k$ vector space spanned by degree 3 monomials in 2 variables, say $x^3, x^2y, xy^2, y^3$. Let $V$ be the sub-vector space spanned by the first 3 symbols; and consider the map induced by the corresponding two-dimensional linear system $L_V$:

$ \phi_V : \mathbb{P}^1 \to \mathbb{P}^2 $

$(x,y) \mapsto (x^3, x^2y, xy^2)=(x^2, xy, x^2)$

This map is easily seen to be an embedding, and hence we say that $L_V$ is very ample. However, $L_V$ has a base point; namely the point $(0,1)$ which is a common zero of every element of $V$.

Now, I'm under the impression that every very ample linear system should be base point free (since being very ample means the associated map is an embedding, which means in particular that it's a morphism, and hence the linear system is base point free.

Cheers, Robert

edit: I seem to have several different accounts - could someone please point me in the right direction as to how I can merge them all?