2016/06/28 by Antonio Macchia, Jorge Neves, Macchia, Antonio +5
Mathematics · #11T55 #13F20 (primary) #14G15 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:11T55 #msc:13F20 #msc:14G15
paper · pdf · doi:10.48550/arxiv.1606.08621
arxiv created 2016/06/28 · arxiv updated 2016/06/29
Let G be a graph obtained by taking r>=2 paths and identifying all first vertices and identifying all the last vertices. We compute the Castelnuovo--Mumford regularity of the quotient S/I(X), where S is the polynomial ring on the edges of G and I(X) is the vanishing ideal of the projective toric subset parameterized by G. The case we consider is the first case where the regularity was unknown, following earlier computations (by several authors) of the regularity when G is a tree, cycle, complete graph or complete bipartite graph, but specially in light of the reduction of the computation of the regularity in the bipartite case to the computation of the regularity of the blocks of G. We also prove new inequalities relating the Castelnuovo--Mumford regularity of S/I(X) with the combinatorial structure of G, for a general graph.