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Extra info for Combinatorial Commutative Algebra (Graduate Texts in Mathematics, Volume 227)

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How many monomial generators does L have for (d, e) = (3, 3)? 22, it is enough to restrict our attention to monomial ideals, since any initial ideal of an N-graded ideal I has a least as many generators in each degree d as I does. 9 on generic initial ideals, it suﬃces to consider only Borel-ﬁxed monomial ideals, as g · I has the same number of generators in each degree as I does. 4. LEX-SEGMENT IDEALS 35 The degrees of the generators of an ideal measure its zeroth Betti numbers. One can also ask which ideals have the worst behavior with respect to the degrees of the higher Betti numbers.

The drawing of Buch(I ) there contains the nonplanar complete bipartite graph K3,3 as a subgraph.

12. We ﬁnd that K(S/J; t, t, t) = = 1 − 3t4 − 3t6 + 3t7 + 9t8 − 7t9 (1 − t)3 · (1 + 3t + 6t2 + 10t3 + 12t4 + 12t5 + 7t6 ). The last factor of degree 6 is the Hilbert series of S/J. A theorem of Bayer and Stillman [BS87] states that the Castelnuovo–Mumford regularity of J can be read oﬀ as the largest degree of a minimal generator of ginrevlex (J). This number is an important invariant, and it equals 7 in our example. Method 3: Resolution by picture. Our main tools for studying monomial ideals in two variables were staircase diagrams.