Moduli spaces is an active field of research in mathematics which play an important role in theoretical physics, and particularly in string theory. This paper focuses on the moduli space of one-dimensional sheaves on the projective plane \(\mathbb{P}^2\) which are studied in [W. Pi and J. Shen, Algebr. Geom. 10, No. 4, 504–520 (2023; Zbl 1528.14016)], which stands as the main reference of the present work. These moduli spaces depends on \(d\) (degree of the sheaf), and the Euler characteristic \(\chi\): for each pair \((d,\chi)\) we have the moduli space \(M_{d,\chi}\) of semistable one-dimensional sheaves on the projective plane. It is proven that \(M_{d,\chi}\) is an irreducible projective variety which is smooth on the stable points. However, for the case \(d\) and \(\chi\) coprimes, the condition of semistability and stability coincide, and then \(M_{d,\chi}\) is non-singular.
The main result is to extend the theorem proved by Woolf which states that \(M_{d,\chi}\) and \(M_{d,\chi'}\) are isomorphic {\itshape as algebraic varieties} if and only if \(\chi\equiv \pm \chi\pmod{d}\) [M. Woolf, “Nef and Effective Cones on the Moduli Space of Torsion Sheaves on the Projective Plane”, Preprint, arXiv:1305.1465]. This result is somewhat expected by the symmetries of \(M_{d,\chi}\), or in other words, it claims that for a given \(d\in \mathbb{Z}\), the unique symmetries are the natural ones given by duality and torsion by \(\mathcal{O}_{\mathbb{P}^2}(1)\). In this reviewed paper, the authors show that this theorem holds if we consider \(M_{d,\chi}\) as topological spaces, that is, \(M_{d,\chi}\) and \(M_{d',\chi'}\) are isomorphic {\itshape as topological spaces} if and only if \(\chi\equiv\pm \chi\pmod{d}\). However, this statement is a neat consequence of another analogous result: if \(A^\ast(M_{d,\chi})\) denotes the even cohomology of \(M_{d,\chi}\), then \(A^\ast(M_{d,\chi})\cong A^\ast(M_{d,\chi'})\) as graded \(\mathbb{C}\)-algebras if and only if \(\chi'\equiv\pm \chi \pmod{d}\). The proof of these theorems goes through the explicit computation of the relations between a set of generators for \(A^\ast(M_{d,\chi})\). Indeed, such explicit computations allows to solve some part of the proofs to a big computation aided with a mathematical software. Indeed, the third author maintains a worksheet available on his website that collects a good part of the calculations used in the demonstration. Finally, we point out the well-structure proof which increase the readability of this paper.