Let $X$ be a projective variety over an algebraically closed field, and $I$ be the homogenous ideal of $X$ and $J$ be an ideal with the same zero set. Suppose that I know $I=\langle f_1,...f_n \rangle$ with $f_i$ of degree $d$, and $J= \langle g_1,...g_m \rangle$ with $g_i$ also of degree $d$. Does it follow that $J$ is also a radical ideal (and hence equal to $I$)?
My motivation: I am trying to show that the quadratic relations (a postiori Plucker relations) that one gets by a certain tensor contraction, generate the ideal of the image of the grassmannian under the Plucker embedding. I know that elements of the coordinate ring of the image can't satisfy any relations of degree 1. So if the criterion above holds then these quadratic relations are actually Plucker relations.