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2 basic comments: 1) $L+H$ big is not equivalent to $(L+H)^3>0$ example: $\mathbb P^2$ blown up at a point, $H=$ any ample and $L=mE$ where $E=$exceptional curve and $m\gg 0$. 2) this is the proo …

By definition, you first define $\sigma_\Gamma(L)$ for big divisors and then you take the limit. In other words, if $L$ is big, then clearly $\sigma_\Gamma(L)$ is non zero for only finitely many divi …

Assume that $X$ is a smooth 3-fold and let $C\subseteq X$ a curve with a unique singular point of multiplicity $2$. Does there exist a smooth surface $S$ inside $X$ which contain $C$ ? Clearly if the …

I am not sure if this is the shortest proof, but I think that it is a proof. Let $A=$ very ample line bundle. After replacing D by a multiple, you may assume that $$C=D - K_X - (n+1) A$$ is ample w …

It is just a consequence of the Riemann-Roch theorem. But another way to see it, is using the fact that if $L$ is any line bundle then $L=A\otimes B^{-1}$ where $A$ and $B$ are very ample line bundles …

I think that the paper Burns-Wahl "Local contributions... " gives some of the examples you are looking for. (sorry I just wanted to add a comment above, but I am not sure how to do it).

What do you mean by stable? usually stable is with respect to a polarization $H$. If $K_X$ is ample and you choose $H=K_X$ then we know by Aubin-Yau's theorem that there exists a Kahler-Einstein metri …