Crepant resolution of \(\mathbf{A}^4 /A_4\) in characteristic 2 (Q6146737)

Let \(G\) be a finite subgroup of \(SL(n, \mathbb{C})\) acting on \(\mathbb{C}^n\). Assume that there exists a crepant resolution \(f: Y\rightarrow \mathbb{C}^n/G\). Batyrev's theorem states that the Euler number of \(Y\) equals the number of conjugacy classes of \(G\). In positive characteristic, Batyrev's theorem holds for non-modular quotient singularities. However, for modular cases, Chen, Du, and Gao provided a crepant resolution of a quotient singularity as a counterexample to Batyrev's theorem in characteristic \(2\). Notably, in their example, the group \(G=C_6\) has non-identity reflections. Now, let \(K\) be an algebraically closed field of characteristic \(2\), and consider \(A_4\), the alternating group acting by permutation on \(\mathbb{A}_K^4\). Denote the quotient singularity \(\mathbb{A}_K^4/A_4\) as \(X\). The author demonstrates that \(X\) possesses a crepant resolution \(\tilde{X}\) with an Euler number of \(10\). Since the alternating group \(A_4\) has \(4\) conjugacy classes, her result is a new counterexample to analogous statement of Batyrev's theorem in positive characteristic, where the group \(G=A_4\) has no non-identity reflections.