(I should preface that I am not good at mathematics. I'm currently struggling through Eisenbud's Commutative Algebra)
When talking about a higher-dimensional, not necessarily smooth, projective algebraic variety $ X $, one can take a projection to a variety of lower dimension $ X' $, find the lines on this new variety, where the set of lines is also a smooth projective variety $ Y $ by the Plücker embedding, and lift these lines in $X ' $ into curves in the original $ X $.
My question is, does the $Aut(Y)$ say anything about the set of curves in $ X $? I want it to be true, in particular allowing for some sort of classification of the curves over $ X $, but I suspect that might be naive.