**Edit**: The formulation of my question was incorrect, for several reasons. Here is what I hope to be the correct formulation:


Let $\mathbb{P}$ be a projective space, and $V$ a *general* linear subspace of $H^0(\mathcal{O}\_{\mathbb{P}}(d))$ (that is, a general point in the corresponding Grassmannian). Then for $p<d$ the multiplication map
$$H^0(\mathcal{O}\_{\mathbb{P}}(p))\otimes V\rightarrow H^0(\mathcal{O}\_{\mathbb{P}}(p+d))$$
is of maximal rank, i.e. either injective or surjective.


Is this true? Known? Sasha's answer shows that it is true when $\dim V \leq \dim \Bbb{P}+1$.



Relations between homogeneous polynomials
I am posting this as an answer since the comment thread is already long. The question is a special case of Fröberg's Conjecture.


MR0813632 (87f:13022)   
 
Fröberg, Ralf(S-STOC)   

An inequality for Hilbert series of graded algebras.   
 
Math. Scand. 56 (1985), no. 2, 117–144.   

13H15 (13D03 13H10)   



This special case is mostly solved by work of Gleb Nenashev.


MR3621254   

Nenashev, Gleb(S-STOC)   

A note on Fröberg's conjecture for forms of equal degrees.   
 
C. R. Math. Acad. Sci. Paris 355 (2017), no. 3, 272–276.   

13D40   

<https://arxiv.org/pdf/1512.04324.pdf>


Theorem 1 proves the maximal rank conjecture for these maps except for a few values of $p$ near the "changeover" from injectivity to surjectivity. In particular, Nenashev proves injectivity whenever $$\text{dim} H^0(\mathbb{P},\mathcal{O}\_{\mathbb{P}}(p))\otimes V \leq \text{dim} H^0(\mathbb{P},\mathcal{O}\_{\mathbb{P}}(p+d)) - \text{dim} H^0(\mathbb{P},\mathcal{O}\_{\mathbb{P}}(p))^2,$$ 
and surjectivity whenever
$$\text{dim} H^0(\mathbb{P},\mathcal{O}\_{\mathbb{P}}(p))\otimes V \geq \text{dim} H^0(\mathbb{P},\mathcal{O}\_{\mathbb{P}}(p+d)) + \text{dim} H^0(\mathbb{P},\mathcal{O}\_{\mathbb{P}}(p))^2.$$ 


