This paper examines the Arithmetically Cohen-Macaulay (ACM) property for certain codimension 2 varieties in P1×P2 called sets of lines in P1×P2 (not necessarily reduced). We discuss some obstacles to finding a general characterization. We then consider certain classes of such curves, and we address two questions. First, when are they themselves ACM? Second, in a non-ACM reduced configuration, is it possible to replace one component of a primary (prime) decomposition by a suitable power (i.e. to “fatten” one line) to make the resulting scheme ACM? Finally, for our classes of such curves, we characterize the locally Cohen-Macaulay property in combinatorial terms by introducing the definition of a fully v-connected configuration. We apply some of our results to give analogous ACM results for sets of lines in P3.

Favacchio G., & Migliore J. (2021). The ACM property for unions of lines in P1×P2. JOURNAL OF PURE AND APPLIED ALGEBRA, 225(11), 106739 [10.1016/j.jpaa.2021.106739].

The ACM property for unions of lines in P1×P2

Favacchio G.;
2021

Abstract

This paper examines the Arithmetically Cohen-Macaulay (ACM) property for certain codimension 2 varieties in P1×P2 called sets of lines in P1×P2 (not necessarily reduced). We discuss some obstacles to finding a general characterization. We then consider certain classes of such curves, and we address two questions. First, when are they themselves ACM? Second, in a non-ACM reduced configuration, is it possible to replace one component of a primary (prime) decomposition by a suitable power (i.e. to “fatten” one line) to make the resulting scheme ACM? Finally, for our classes of such curves, we characterize the locally Cohen-Macaulay property in combinatorial terms by introducing the definition of a fully v-connected configuration. We apply some of our results to give analogous ACM results for sets of lines in P3.
Favacchio G., & Migliore J. (2021). The ACM property for unions of lines in P1×P2. JOURNAL OF PURE AND APPLIED ALGEBRA, 225(11), 106739 [10.1016/j.jpaa.2021.106739].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10447/534047
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