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Let $S_d$ be the vector space of homogeneous polynomials of degree $d$ in two variables $x,y$ over an algebraically closed field $k$. Let $\phi \in S_d^*$ be a linear functional on $S_d$, such that $\phi(F)=0$, for every $F \in S_d$ that is divisible by the polynomial $(x-a y)^2$.

Question: How can we see that $\phi$ is inside the closure of linear functionals of the form $\phi'(F) = \beta \, F(p) + \gamma \, F(q)$, where $\beta,\gamma$ vary over $k$ (but they are not allowed to be zero at the same time), $q$ varies over $k^2-\left\{(0,0)\right\}$ and $p = (a,1)$?

Edit: Closure here refers to the Zariski closure of that set of functionals: any non-zero linear functional on $S^d$ can be viewed as a point of $\mathbb{P}^d$.

Reference: Harris, Algebraic Geometry: A First Course, last sentence at p. 105 of the proof of Proposition 9.7.

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  • $\begingroup$ Did you try to work this out by hand for small values of $d$, say $d=2$ and $d=3$? $\endgroup$ – Jason Starr Jun 23 '15 at 19:11
  • $\begingroup$ @JasonStarr: I am not sure how to approach this irrespectively of $d$. $\endgroup$ – Manos Jun 23 '15 at 20:11

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