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I recently was looking at Chapter 1, Section 4.1 of the book on Cox Rings by Ivan Arzhantsev, Ulrich Derenthal, Jurgen Hausen, and Antonio Laface (see https://arxiv.org/pdf/1003.4229.pdf) I noticed that they assumed that the schemes they study are normal, $ \mathbb{Q} $-factorial, pre-varieties.

In the construction 4.1.1 I see that regularity in codimension one is needed because this is needed to even define the class group. If one works with a variety and not just a pre-variety, then by Chapter II, Schemes, Section 6, Divisors, Proposition 6.15 the Cartier Class group is isomorphic to the Picard group. The isomorphism of the Cartier Class group and Picard group, and $ \mathbb{Q} $-factoriality seem to show that $ \mathcal{O}(D_{1}) \cdot \mathcal{O}(D_{2}) \cong \mathcal{O}(D_{1}+D_{2}) $. This seems to be the only other needed ingredient for their construction to work.

However, I do not see any reason why normality is needed to define the Cox ring of a $ \mathbb{Q} $-factorial, variety. The only reason I see normality included is to obtain regularity in codimension one. Is there a reason why the authors include normality, or is the Cox ring well defined in this case but other desired properties do not hold?

Schemer1
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1 Answers1

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The issue might be elsewhere than you are expecting. Perhaps the relevant question is this: How do you define a divisor on a non-normal variety? (See https://mathoverflow.net/a/46663/10076 for some of the issues you run into).

The best way to handle this would be to work with divisorial sheaves (reflexive sheaves of rank 1) instead of divisors, so essentially just the "$\mathscr O(D)$' without the "$D$", but then the multiplication is a bit tricky as you would have to take the reflexive hull of the product. This would probably work, but perhaps the authors didn't want to write another chapter on divisorial sheaves and their arithmetic on non-normal varieties.

Sándor Kovács
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  • Thank you. I forgot to mention that I use Hartshorne's convention that a variety is an integral, separated, scheme, of finite type over an algebraically closed field. The authors of this book did not. – Schemer1 Aug 17 '22 at 20:40
  • One question I have is the following. In your response to Jesus Martinez Garcia's question in https://mathoverflow.net/a/46663/10076 you mentioned that $ S_{2} $ allowed one to assume that if $ \operatorname{codim}(Z) \ge 2 $, then if $ f $ is nowhere zero on $ X \setminus Z $, it is nowhere zero on $ X $. If $ X $ is $ \mathbb{Q} $-Cartier, then $ \operatorname{div}(f^{m}) $ is a principal Cartier divisor for some $ m \in \mathbb{N} $. As a result multiples of principle Weil divisors correspond to principle Cartier divisors. If we do not require that the vanishing locus of a (cont). – Schemer1 Aug 17 '22 at 21:00
  • ...regular function be a codimension one sub-variety, but rather define the divisor of a regular function $ f $ in the class group to be $ \sum_{Y} v_{Y}(f) $ where $ Y $ is a prime Weil divisor as Hartshorne does, then it seems the Cox ring of $ X $ is well defined, but that an element $ f \in \operatorname{Cox}(X) $ may vanish on sub-varieties of codimension two. Is this correct? Thank you Sandor for your referral to your answer to Jesus Martinez Garcia's question, and your answer to mine. – Schemer1 Aug 17 '22 at 21:05
  • Yes, that could happen (a regular function vanishing only on a codimension two subset). – Sándor Kovács Aug 18 '22 at 23:08