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Plano
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Plano Asks: Different definition of Cox rings
Definition: Let $X$ be a normal projective variety with finitely generated Picard group. Define the Cox ring of $X$ as the multisection ring $$\text{Cox}(X)=\bigoplus_{(m_1,\ldots,m_k)\in \mathbb{N}^k} \text{H}^0(X,m_1L_1+\ldots+m_kL_k),$$ where $L_1,\ldots,L_k$ are a basis of $\text{Pic}(X)_{\mathbb{Q}}$ and whose affine hull contains $\overline{\text{Eff}(X)}$.
This is the Hu-Keel defintion of Cox ring, and I would like to understand why the second extra property is required, since by looking at the literature it looks like there are very few cases in which this condition is asked, and there is no explanation to that. In particular, these are my doubts:
I apologize in advance for this low-level (and probably not research-oriented) question, I've asked the same question on MSE without receving an proper answer (I've then deleted since they were equal, and here there are some comments), thus I understand if you want to delete it. Thanks in advance!
Definition: Let $X$ be a normal projective variety with finitely generated Picard group. Define the Cox ring of $X$ as the multisection ring $$\text{Cox}(X)=\bigoplus_{(m_1,\ldots,m_k)\in \mathbb{N}^k} \text{H}^0(X,m_1L_1+\ldots+m_kL_k),$$ where $L_1,\ldots,L_k$ are a basis of $\text{Pic}(X)_{\mathbb{Q}}$ and whose affine hull contains $\overline{\text{Eff}(X)}$.
This is the Hu-Keel defintion of Cox ring, and I would like to understand why the second extra property is required, since by looking at the literature it looks like there are very few cases in which this condition is asked, and there is no explanation to that. In particular, these are my doubts:
- Why do they add this condition: I know it's vague, and it is a definition so it is not correct or wrong a priori, but for istance can we always find such a basis?
- What do they mean by affine hull: I suspect they mean the convex hull, but these are two different notions.
I apologize in advance for this low-level (and probably not research-oriented) question, I've asked the same question on MSE without receving an proper answer (I've then deleted since they were equal, and here there are some comments), thus I understand if you want to delete it. Thanks in advance!
^2(dn^2 - dx^2)$ where $dn$ is the conformal time differential form. The Klein Gordon equation in curved spacetime is $(\frac{1}{g^{1/2}}\partial_{\mu}(g^{1/2}g^{\mu\nu}\partial_{\nu}) + m^2)\phi = 0$
