I was looking at point (I) of exercise 13.14 form Eisenbud commutative algebra with a view toward algebraic geometry.
The problem is:
Given two projective varieties $X,Y$ in $\mathbb{P}^r(k)$ (here $k$ is an algebrically closed field) with homogeneous coordinate ring $R_X,R_Y$;
let $J(X,Y)$ be the projective variety defined by all the lines that connect a point of $X$ and a point of $Y$ (where $X$ and $Y$ are both embedded in $\mathbb{P}^{2r+1}$).
Show that a coordinate ring for $J(X,Y)$ is the ring $R:=R_X\otimes_k R_Y$ graded by $R_d:=\oplus_{i+j=d}{R_X}_i\otimes_k {R_Y}_j$.
I don't know how to solve this problem, i tried to find the ideal that define $J(X,Y)$ and than show that there is an isomorphism. I have troubles in finding a definition for that ideal.
Is there another approach? Can i find a definition for that ideal?
Suggestions and hints are welcome.