2020/09/07 by Giuseppe Favacchio, Favacchio, Giuseppe, Juan Migliore +1
Computer Science · Mathematics · #13A15 #13H10 #14M05 #14N20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2009.03246
openalex publication_date 2020/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper examines the Arithmetically Cohen-Macaulay (ACM) property for certain codimension 2 varieties in \mathbb P1× \mathbb P2 called sets of lines in \mathbb P1× \mathbb 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 \mathbb P3.