The information encoded in initial ideals (Q2701664)

Let \(I\) denote a homogeneous ideal in a polynomial ring \(S=k[x_0,x_1,\dots,x_r]\). After fixing a monomial order, one may consider the initial ideal in\((I)\). It is an interesting question to determine how much information about \(I\) may be gleaned from in\((I)\). The authors consider this problem for the reverse lexicographic order. (This monomial order has certain nice properties, such as in\((I:x_r)=\) in\((I):x_r\).) First, the authors define a sequence of ideals \(I^j\) (not the \(j\)-th power of \(I\)) by NEWLINE\[NEWLINEI^j=\text{im}((I:x_r^j)\rightarrow S\rightarrow S/(x_r)).NEWLINE\]NEWLINE The Hilbert functions of the \(I^j\) constitute a more refined invariant of \(I\) than the Hilbert function of \(I\) and are closely related to in\((I)\). Using these algebraic preliminaries, the authors go on to consider the information (mainly of a cohomological nature) encoded in the generic initial ideal in several geometric examples, such as space curves, zero-dimensional subschemes of 3-space, and the intersection of a curve and a surface in 3-space. NEWLINENEWLINENEWLINEIn the last section, they generalize these ideas to the generic higher initial ideal gin\(_1(I)\) in the case of a space curve \(C\). As a consequence, it is shown that if \(C\) is linked to a curve \(X\), then knowing the generic higher initial ideal of \(I_C\) is equivalent to knowing the generic initial ideal of \(I_X\). Higher initial ideals were introduced previously by the first author [\textit{G. Fløystad}, ``Higher initial ideals of homogeneous ideals'', Mem. Am. Math. Soc. 638 (1998; Zbl 0931.13013)], while the second author has previously written an expository article on generic initial ideals [\textit{M. L. Green}, in: Six lectures on commutative algebra. Lect. Summer School, Bellaterra 1996, Prog. Math. 166, 119-186 (1998; Zbl 0933.13002)].