We show the compactly supported motive of the moduli stack of degree n rational curves on the weighted projective stack \({\mathcal {P}}(a,b)\) is of mixed Tate type over any base field K with \(\hbox {char}(K) \not \mid a,b\) and has class \({\mathbb {L}}^{(a+b)n+1}-{\mathbb {L}}^{(a+b)n-1}\) in the Grothendieck ring of stacks. In particular, this improves upon the results of (Han and Park in Math Ann 375(3–4), 1745–1760, 2019) regarding the arithmetic invariant of the moduli stack \({\mathcal {L}}_{1,12n} :=\mathrm {Hom}_{n}({\mathbb {P}}^1, \overline{{\mathcal {M}}}_{1,1})\) of stable elliptic fibrations over \({\mathbb {P}}^{1}\) with 12n nodal singular fibers and a marked Weierstrass section.

我们显示程度的模堆的紧支撑动机Ñ有理曲线的加权投影堆栈上\（{\ mathcal {P}}（A，B）\）是混合泰特型过任何基本字段的ķ与\（\ hbox {char}（K）\ not \ mid a，b \）并具有类\（{\ mathbb {L}} ^ {（a + b）n + 1}-{\ mathbb {L}} ^ {（ a + b）n-1} \）在堆栈的Grothendieck环中。特别是，这改善了（Han和Park在Math Ann 375（3-4），1745-1760，2019）中关于模堆栈\（{\ mathcal {L}} _ {1， 12n}：= \ mathrm {Hom} _ {n}（{\ mathbb {P}} ^ 1，\ overline {{\ mathcal {M}}} _ {1,1}）\）稳定的椭圆形纤维在\ （{\ mathbb {P}} ^ {1} \）与12 n 节点奇异纤维和明显的Weierstrass截面。