The virtual resolutions package for Macaulay2 (Q2040896)

This paper introduces the \textit{Macaulay2} package \texttt{VirtualResolutions}. This package provides some methods to compute \textit{virtual resolutions} of particular subvarieties of smooth projective toric varieties. In some cases, a virtual resolution represents a quick alternative way (beyond the minimal graded free resolutions) to study algebraic invariants of toric subvarieties. Moreover, the package contains methods for constructing curves in \(\mathbb{P}^1\times \mathbb{P}^2\). The latter are source of interesting examples of virtual resolutions. More in detail, the authors of the package have implemented some constructions given by Berkesch, Erman and Smith [\textit{C. Berkesch} et al., Algebr. Geom. 7, No. 4, 460--481 (2020; Zbl 1460.14021)]: \begin{itemize} \item[-] \texttt{virtualOfPair} creates a virtual resolution from a finitely generated module over multigraded ring (or free resolution) and a multidegree; \item[-] \texttt{resolveViaFatPoint} creates a virtual resolution of a zero-dimensional scheme; \item[-] \texttt{isVirtual} checks whether a chain complex is a virtual resolution. \end{itemize} Furthermore, some functions are devoted to create the ideal of a curve in \(\mathbb{P}^1\times \mathbb{P}^2\) following some different approaches. For instance, one can use the functions \texttt{curveFromP3toP1P2}, \texttt{randomCurveP1P2}, \texttt{randomMonomialCurve} or \texttt{randomRationalCurve} according to specific needs. All descriptions in the paper are also accompanied with exhaustive examples of how to use the \textit{Macaulay2} instructions to solve the problem in question. A technical description of the package, with all functions and commands, can be found on the following website: \url{https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.17/share/doc/Macaulay2/VirtualResolutions/html/index.html}.