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S. Carnahan ♦
S. Carnahan ♦
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Option 1 is definitely not the definition - in the scheme setting, it doesn't yield log structures on the divisors in question, and you don't get log points.

Option 2 works A.J. mentioned that you probably meant to say, "stack equipped with a log structure". You can find a definition in section 5 of Martin Olsson's paper Logarithmic geometry and algebraic stacks. His web page doesn't have it, but you can find it on Google ScholarGoogle Scholar.

Option 2 works also. Fine log schemes over a base fine log scheme form an algebraic stack over the underlying base scheme. YouYou can do your log stack theory over this. See Corollary 5.8 in Olsson's paper comparing the two options.

Option 1 is definitely not the definition - in the scheme setting, it doesn't yield log structures on the divisors in question, and you don't get log points.

Option 2 works. You can find a definition in Martin Olsson's paper Logarithmic geometry and algebraic stacks. His web page doesn't have it, but you can find it on Google Scholar. Fine log schemes over a base fine log scheme form an algebraic stack over the underlying base scheme. You can do your log stack theory over this.

Option 1 is definitely not the definition - in the scheme setting, it doesn't yield log structures on the divisors in question, and you don't get log points. A.J. mentioned that you probably meant to say, "stack equipped with a log structure". You can find a definition in section 5 of Martin Olsson's paper Logarithmic geometry and algebraic stacks. His web page doesn't have it, but you can find it on Google Scholar.

Option 2 works also. Fine log schemes over a base fine log scheme form an algebraic stack over the underlying base scheme. You can do your log stack theory over this. See Corollary 5.8 in Olsson's paper comparing the two options.

clarification
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S. Carnahan ♦
S. Carnahan ♦
  • 45.7k
  • 6
  • 114
  • 220

Option 1 is definitely not the definition - in the scheme setting, it doesn't yield log structures on the divisors in question, and you don't get log points.

Option 2 works. You can find a definition in Martin Olsson's paper Logarithmic geometry and algebraic stacks. His web page doesn't have it, but you can find it on Google Scholar. Fine log schemes over a base fine log scheme form an algebraic stack over the underlying base scheme. You can do your log stack theory over this.

Option 1 is definitely not the definition - in the scheme setting, it doesn't yield log structures on the divisors in question, and you don't get log points.

Option 2 works. You can find a definition in Martin Olsson's paper Logarithmic geometry and algebraic stacks. His web page doesn't have it, but you can find it on Google Scholar. Fine log schemes over a base fine log scheme form an algebraic stack. You can do your log stack theory over this.

Option 1 is definitely not the definition - in the scheme setting, it doesn't yield log structures on the divisors in question, and you don't get log points.

Option 2 works. You can find a definition in Martin Olsson's paper Logarithmic geometry and algebraic stacks. His web page doesn't have it, but you can find it on Google Scholar. Fine log schemes over a base fine log scheme form an algebraic stack over the underlying base scheme. You can do your log stack theory over this.

Source Link
S. Carnahan ♦
S. Carnahan ♦
  • 45.7k
  • 6
  • 114
  • 220

Option 1 is definitely not the definition - in the scheme setting, it doesn't yield log structures on the divisors in question, and you don't get log points.

Option 2 works. You can find a definition in Martin Olsson's paper Logarithmic geometry and algebraic stacks. His web page doesn't have it, but you can find it on Google Scholar. Fine log schemes over a base fine log scheme form an algebraic stack. You can do your log stack theory over this.