Discriminants of cubic curves and determinantal representations (Q2084549)

In the article under review, the author provides a new proof of the formula for the discriminant of a smooth plane cubic curve over \(\mathbb{C}\) in terms of theta constants using determinantal representations of the curve. Firstly, the author considers the case of a smooth curve in the Weierstrass form \[ \varphi(x,y,z) = y^2z - 4x^3 + g_2xz^2 + g_3z^3. \] They compute the determinantal representations of \(\varphi\) over a general field \(K\) and give a particular representation for \(K = \mathbb{C}\). Then, using the determinantal representation together with a formula relating discriminant and certain resultant, they prove the classical formula \[ \Delta_{\varphi} = 2^{16}\left(\frac{\pi}{\omega_1}\right)^{12}(abc)^8, \] where \(a, b, c\) are even theta constants. A similar strategy works for smooth plane cubic curves. The author provides determinantal representations for any non-rational complex plane curve. Then the choice of a sufficiently nice representation in the case of smooth plane cubics over \(\mathbb{C}\) yields the formula \[ \Delta_{\varphi} = \frac{2^{16}}{\det(M)^{12}}\left(\frac{\pi}{\omega_1}\right)^{12}(abc)^8, \] where \(M\) is a matrix of a linear change of coordinates transforming the polynomial \(\varphi\) to a Weierstrass form.