We give a proper definition of the multiplicative structure of the following rings: the Cox ring of invertible sheaves on a general algebraic stack and the Cox ring of rank one reflexive sheaves on a normal and excellent algebraic stack. We show that such Cox rings always exist and establish their (non-)uniqueness in terms of an Ext-group. Moreover, we compare this definition with the classical construction of a Cox ring on a variety. Finally, we give an application to the theory of Mori dream stacks.
COX RINGS OF ALGEBRAIC STACKS
Elena Martinengo;
2023-01-01
Abstract
We give a proper definition of the multiplicative structure of the following rings: the Cox ring of invertible sheaves on a general algebraic stack and the Cox ring of rank one reflexive sheaves on a normal and excellent algebraic stack. We show that such Cox rings always exist and establish their (non-)uniqueness in terms of an Ext-group. Moreover, we compare this definition with the classical construction of a Cox ring on a variety. Finally, we give an application to the theory of Mori dream stacks.
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