On the generators of the first homology with compact supports of the Weierstrass family in characteristic zero

Abstract:

Let ${{\mathbf {W}}_{\mathbf {Q}}} = \operatorname {Proj}({\mathbf {Q}}[{{\text {g}}_2},{g_3},X,Y,Z]/({\text {homogeneous ideal generated by }} - {Y^2}Z + 4 {X^3} - {g_2} X{Z^2} - {g_3} {Z^3}))$. This is said to be the Weierstrass Family over the field ${\mathbf {Q}}$. Then the first homology with compact supports of the Weierstrass Family is computed explicitly, i.e., it is generated by ${\{ {C^{ - k}} dX \wedge \;dY\}_{k \geqslant 1}}$ and ${\{ X{C^{ - k}}dX \wedge dY\}_{k \geqslant 1}}$ over the ring ${\mathbf {Q}}[{g_{2}},{g_3}]$, where $C$ is a polynomial ${Y^2} - 4{X^3} + {g_2}X + {g_3}$. When one tensors the homology of the Weierstrass Family with ${\Delta ^{ - 1}} {\mathbf {Q}}[{g_2},{g_3}]$, being localized at the discriminant $\Delta = g_2^3 - 27g_3^2$, over ${\mathbf {Q}}[{{\text {g}}_2},{g_3}]$, the first homology is generated by ${C^{ - 1}}dX\; \wedge \;dY$ and $X{C^{ - 1}}dX \wedge dY$. One also obtains the first homologies with compact supports of singular fibres over $\wp = ({g_2} = {g_{3}} = 0)$ and $\wp = ({g_2} = 3,{g_{3}} = 1)$ as corollaries.