Celledoni et al [1,2] have illustrated the power of using so-called Darboux polynomials to search for preserved integrals and measures of a rational map L (including a map arising as a Kahan-Hirota-Kimura discretisation of a (Hamiltonian) ODE). For instance, integrals are obtained as the ratio of two Darboux polynomials where each Darboux polynomial satisfies an equation of the form P(x') = C(x) P(x), where C(x) is a rational function called a cofactor (and prime denotes the image of x under L). In [1,2], C(x) is assumed to take a form involving factors from the numerator and denominator of the Jacobian determinant of L. In this talk, we confirm this "Jacobian factor ansatz" for C(x) using concepts from algebraic geometry and explore its further consequences. [1] E.Celledoni, C.Evripidou, D.I.McLaren, B.Owren, G.R.W. Quispel and B.K.Tapley, Detecting and determining preserved measures and integrals of birational maps, J. Comput. Dyn. 9 (2022), no. 4, 553–57 [2] E.Celledoni, C.Evripidou, D.I.McLaren, B.Owren, G.R.W. Quispel, B.K.Tapley and P.H. van der Kamp, Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps, J. Phys. A 52 (2019), 11pp.