In this paper, building on their previous work with T. Yasuda, the authors prove a precise version of inversion of adjunction and the semicontinuity of minimal log discrepancies for local complete intersection varieties. This extends the results for smooth varieties given in [Invent. Math. 153, 519–535 (2003; Zbl 1049.14008)].
If \(X\) is a \(\mathbb Q\)-Gorenstein normal variety, \(Y=\sum _{i=1}^k q_i\cdot Y_i\) a formal combination of proper closed subschemes \(Y_i\) with coefficients \(q_i\in \mathbb{R} _+\) and \(W\subset X\) is a closed subset, then the minimal log discrepancy of \((X,Y)\) along \(W\) is given by \(\text{mld}(W;X,Y):=\min _{c_X(E)\subset W} a(E;X,Y)\) where \(E\) is any prime divisor on a smooth variety \(X'\) with a proper birational morphism \(f:X'\to X\), \(c_X(E)=f(E)\) is its center and \(a(E;X,Y)-1\) is the coefficient of \(E\) in \(K_{X'/X}-f^{-1}(Y)\).
Assuming that \(X\) is a normal local complete intersection variety, the authors show that:
1) If \(D\subset X\) is a normal effective Cartier divisor such that \(D\not\subset \bigcup Y_i\), then for every proper closed subset \(W\subset D\), we have \(\text{mld}(W;X,Y+D)=\text{mld}(W;D,Y| _D)\).
2) The function \(x\to \text{mld}(x;X,Y)\) is lower semicontinuous.
3) \(X\) has log canonical (resp. canonical, terminal) singularities if and only if its \(m\)-th jet scheme \(X_m\) is equidimensional (resp. irreducible, normal) for all \(m\).