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I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf . Here, a toric variety has ...

A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold. The complex manifold $X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ admits a non-trivial line bundle for the ...

Let $X_{\Sigma}$ be a projective toric variety. Consider the total coordinate ring $$S = \mathbb{C}[x_{\rho}\: | \: \rho\in\Sigma(1)]$$ Define $\deg(x_{\rho}) = D_{\rho}$. Now, take a divisor $D = \...

In https://arxiv.org/pdf/hep-th/0005247.pdf, on page 60 and 61, it is mentioned that the connection of $\mathcal{O}(-n)$ over a (simplicial) toric variety of the form $$ (\mathbb{C}^N \backslash U)/(\...

Question: Let $n$ be a positive integer and let $X$ be a simplicial toric variety. Does every coset of $n\cdot Pic(X)\subseteq Pic(X)$ contain a base point free linear system? If $X$ is not ...