# Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem.

Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without base point. Then $\phi$ is an embedding iff the following is satisfied:
(a) For all pairs $p,q\in X$ the subspace $V(-p-q)$ of sections vanishing at both $p,q$ has dimension dim$V-2$, and
(b) $V(-2p)$ is properly contained in $V(-p)$ for each $p\in X$.

I can see that condition (a) implies set-theoretical injectivity.

The lecture says (b) is to ensure $\phi$ is an immersion because then at each point $p$ there is a function in $V$ that vanishes at $p$, but does not vanish to order two. But I'm not satisfied because I can't write this argument down in detail. How does one prove surjectivity on the level of stalk given condition (b), so that $\phi$ is an immersion. Thank you very much.

For a general, say, normal variety $X$, does a linear system with condition (b) define a map $\phi$ which is an immersion ($p$ replaced by a divisor in the system)? Thank you.

• There is a proof in Hartshorne II.7. In general the condition is that $V \otimes \mathfrak m_p^2$ is properly contained in $V \otimes \mathfrak m_p$. – user64687 Oct 20 '14 at 7:55