Let $C$ be a reduced, projective curve and denote by $\mathbb{C}_x$ the skyscreaper sheaf of lenght one supported at $x$. Is there an easy way to compute
$Hom(\mathbb{C}_x,\mathbb{C}_y$), $Ext^1(\mathbb{C}_x,\mathbb{C}_y)$ and $Hom(\mathbb{C}_x,V)$, where $V$ is a vector bundle over $C$ and $y\in C$?