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Just address the question as to whether one can read off the fact that $\tfrac1r(a_1,\ldots,a_n)$ is canonical from the numbers $r$ and $a_1,\ldots,a_n$: the answer is yes. It is determined by the Rei …

If $X=\mathbb{V}(I)$ is given by the ideal $I$, then the $m$th symbolic power $I^{[m]}$ consists of all those functions vanishing to multiplicity $m$ at the generic point of $X$. Thus a hypersurface $ …

If $n=5$ then let $\mathbb P^5$ have coordinates $x_0,\ldots,x_5$ and suppose the plane is $H=\mathbb P^2_{(x_0:x_1:x_2)}$. The two equations of $X$ are necessarily of the form $$ \begin{pmatrix} A_1 …

Yes - there are very many such examples, and you can cook up examples by a procedure called 'Mori's algorithm'. A k2A flipping neighbourhood is a 3-fold flipping contraction $f\colon(C\subset X)\to (P …

You are using the wrong equation to compute discrepancies. It should be $$ K_Y = f^*(K_X + D) + \sum a_E(X,D) E $$ where the $E$ are not all necessarily exceptional. For example if $(X,D)$ is already …