Borel degenerations of arithmetically Cohen-Macaulay curves in \(\mathbb{P}^{3}\)
Consider an algebraically closed field \(k\) of characteristic zero, the projective space \(\mathbb{P}^n = \mathrm{Proj}(S:=k[x_0,\dots,x_n])\) and a numerical polynomial \(p(t) \in \mathbb{Q}[t]\). The Hilbert scheme \(\mathcal{H}\mathrm{ilb}_{p(t)}(\mathbb{P}^n)\) is a projective scheme that parameterizes all closed subschemes in \(\mathbb{P}^n\) whose Hilbert polynomial is \(p(t)\). The understanding of the component structure of a Hilbert scheme is in general very difficult. A major tool to tackle this problem is the study of the class of Borel-fixed ideals in \(k[x_0,\dots,x_n]\). These ideals define the most degenerate objects on the Hilbert scheme, their orbits under the action of \(\mathrm{GL}(n+1)\) are closed and they have nice combinatorial properties. In fact, Borel-fixed ideals have been used in this context since the proof of the connectedness of the Hilbert scheme [\textit{R. Hartshorne}, Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)].NEWLINENEWLINEIn the paper under review, the authors investigate Borel-fixed ideals lying on components of the Hilbert scheme parameterizing arithmetically Cohen-Macaulay schemes of codimension two in \(\mathbb{P}^n\). These components are in 1-to-1 correspondence with Borel-fixed ideals NEWLINE\[NEWLINEJ(a,\boldsymbol{b}) = (x_0^a,x_0^{a-1}x_1^{b_1},\dots,x_0 x_1^{b_{a-1}},x_1^{b_a}) \subset k[x_0,\dots,x_n],NEWLINE\]NEWLINE where \(a \in \mathbb{N}\) and \(\boldsymbol{b}=(b_0,b_1,\dots,b_a) \in \mathbb{N}^{a+1}\) with \(0=b_0<b_1 < \cdots< b_a\). The authors give two necessary conditions for a saturated Borel-fixed ideal \(J\) to define a point on the component corresponding to \(J(a,\boldsymbol{b})\) and they conjecture that this two conditions are also sufficient. The first condition is a consequence of the semi-continuity of ranks of cohomology groups on the fibers of a flat family and says that if a saturated ideal \(J\) defines a point on the component of \(J(a,\boldsymbol{b})\), then \(\dim_k (S_d/J_d) \leq \dim_k (S_d/J(a,\boldsymbol{b})_d)\), for all \(d\). The second condition proved in the paper says that if a saturated Borel-fixed ideal \(J\) defines a point on the component of \(J(a,\boldsymbol{b})\), then the monomials \(x_0^{a-s} x_1^{b_0+\cdots+b_s}\), for each \(s=0,\dots,a\), are contained in \(J\).NEWLINENEWLINEIn the second part of the paper, the authors study in details the case of ACM curves in the projective space \(\mathbb{P}^3\). They present many interesting example to support their conjecture and they prove it in some special case. The main technique used in the proofs is to consider deformations of the ideal \(J(a,\boldsymbol{b})\) and then to determine Gr\"obner degenerations according to different term orderings.
scientific article; zbMATH DE number 6356171