I am currently studying singularities in Algebraic Geometry and wanted to understand why the rank of the Jacobian matrix would characterise a point of singularity/non-singularity (assuming we start off with singularities through some geometric motivation)? What is the reason behind looking at the rank of the Jacobian matrix? Why should a point whose Jacobian matrix has rank $n-r$ be a non-singular point?
Since in Algebraic Geometry, we are working with arbitrary [algebraically closed] fields, it will be hard to visualise, but a differential geometric explanation will also do.
Thanks a lot in advance!
Let me upgrade my comment in to an answer.
For an variety $X\subset \Bbb A^n_k$ defined over a field $k$, the tangent space $T_pX$ at a point $p\in X$ is naturally a subspace of the tangent space $T_p\Bbb A^n_k$. This subspace is exactly the kernel of the Jacobian matrix, so if the Jacobian matrix is of rank $n−r$, then the tangent space is of dimension $r$ - if the variety $X$ is of dimension $r$ at $p$, then this means that $X$ is nonsingular at $p$.