Let $ L \subset \mathbb{P}^n $ be a linear subspace of dimension $ n-1 $, be $ X \subset L $ irreducible projective subvariety and $ y_0 \in \mathbb{P}^n - L $. $ \forall x \in X $ let's consider $ L_x $ the line between $ x $ and $ y_0 $, consider the set $$ Y = \cup_{x \in X} L_x $$ Show that $ Y $ is irreducible projective variety and $ dimY = dimX + 1 $
In this case, the linear subspace of a projective refers to: Consider $ V \subset K^{n + 1} $ linear subspace of dimension $ n $, then $ \mathbb{P}(V) \subset \mathbb{P}^n $ is linear subspace of dimension $ n-1 $
I already showed that it is irreducible, I need to calculate what are the equations that define it as a variety, any suggestions?
Hint: Choose coordinates on $\mathbb{P}^n$ so that $L=(x_n=0)$ and $y_0=(0:0:\dots:0:1)$. The defining equations of $X\subset\mathbb{P}^{n-1}$ then gives ...