# Weighted Projective Spaces and varieties

Consider the weighted projective space $\mathbb{P}(1,1,2)$ with variables $x_0,x_1,x_2$ of degrees 1,1,2 respectively. Consider the map $$(a_0,a_1,a_2) \to (a_0^2,a_0 a_1,a_1^2,a_2)$$ with $F:\mathbb{P}(1,1,2) \to \mathbb{P}^{3}$.

This should be a rather trivial question, but to show that this is defined by the equation $$y_0 y_2 - y_1^2 = 0$$ would it suffice to show that the above image simplifies under definition of projective space to $(a_0,a_1,\frac{a_{1}^{2}}{a_{0}},\frac{a_2}{a_0})$ from which the above equation follows? Why do we not need the equation for the surface defined by $y_0 y_3 = y_2$?

• Do yo want to show that the image is defined by the equation $y_0y_2 -y_1^2=0$? I don't follow the question regarding $y_0y_3-y_2=0$. – baharampuri May 27 '15 at 3:28
• Yes, I guess I proceeded incorrectly by assuming that I could simply find the equations by equating the corresponding components? With $(y_{0},y_{1},y_{2},y_{3}) = (y_{0},y_{1},\frac{y_{1}^{2}}{y_{0}}, \frac{y_{2}}{y_{0}})$? – user238194 May 27 '15 at 4:42