---

Source. Proposition 4.3, p. 16, with Example 4.4 (p. 17) and the example after Definition 4.1 (p. 15), of Serkan Hoşten and Diane Maclagan, The vertex ideal of a lattice, arXiv:math/0012197v1 (2000), published in Adv. in Appl. Math. 29 (2002), 521--538, as identified on the source card. Page numbers are those of the arXiv print.

Statement

Proposition 4.3 (p. 16). If $\mathcal L$ is a two-dimensional lattice in $\mathbf Z^2$, then $P_{\mathcal L}=V_{\mathcal L}$, where $P_{\mathcal L}$ is the product ideal.

The hypothesis that the ambient lattice is $\mathbf Z^2$ is needed. The example after Definition 4.1 ($A=[3\ 4\ 5]$, p. 15) is a two-dimensional lattice in $\mathbf Z^3$ with $P_{\mathcal L}\ne V_{\mathcal L}$ (p. 17), and Example 4.4 (p. 17) is a three-dimensional lattice in $\mathbf Z^3$ with $P_{\mathcal L}$ strictly contained in $V_{\mathcal L}$, computed with Macaulay2.

Read depth. Claims checked: the statement was read clause by clause on p. 16, with the proof; the ideals of Example 4.4 were not recomputed.

Proof pointer

Page 16. Suppose $x^uy^v\in V_{\mathcal L}\setminus P_{\mathcal L}$. A case analysis on the vertices of the planar fiber $P_{(u,v)}$, using the line through $(u,v)$ and a suitable vertex, shows in every case that $x^uy^v$ lies in $P_{\mathcal L}$ after all.

Dependencies

Definition 4.1.

Bears on

No problem page of this corpus.