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Then, I have heard as well that for ANY algebraic variety such that the canonical bundle is defined: $$\mathcal{K}=\mathcal{O}_{X,-\sum D_i}$$ where the $D_i$ are representatives of all divisors in the … In the case of toric varieties, $\sum D_i$~0 if all the primitive generators for the divisors lie on a hyperplane. Then the sum is 0 and therefore the toric variety is Calabi-Yau. …

Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have $$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$ This is the famous Riem …

Suppose $S\subset \mathbb{P}^n$ is a smooth del Pezzo surface and $C$ is an irreducible smooth curve (you can make it rational if it simplifies the setting) such that $\mathcal{L}=\vert -K_S-C\vert $ …

This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow A^2/\m …

Question: Are there methods to find information (dimension, base points, incidence) about linear systems of divisors in a surface given (some) explicit information about the geometry of that surface? …

A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$. Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. The …

I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. … In particular I would like to see examples of linear systems of divisors and how given a linear system of dimension $n$ I can choose a pencil inside it. …