Splayed divisors and their Chern classes (Q2854255)

Conditions for splayedness are investigated. Two divisors in a nonsingular variety \(V\) are splayed at a point \(p\) if their local equations at \(p\) may be written in terms of disjoint sets of analytic coordinates. The paper is a continuation of [\textit{E. Faber}, Publ. Res. Inst. Math. Sci. 49, No. 3, 393--412 (2013; Zbl 1277.32032)] with application to Chern-Schwartz-MacPherson classes redefined in [\textit{P. Aluffi}, C. R., Math., Acad. Sci. Paris 342, No. 6, 405--410 (2006; Zbl 1085.14006)]. For example it is show that if locally \(D_1=(f_1)\) and \(D_2=(f_2)\) are two divisors, then \(D_1\) and \(D_2\) are splayed if and only if NEWLINE\[NEWLINE\frac{J'_{f_1}}{(f_1)}\oplus \frac{J'_{f_2}}{(f_2)}=\frac{J'_{f_1f_2}}{(f_1f_2)}\,,NEWLINE\]NEWLINE where \(J'_{g}\) denotes the ideal generated by \(g\) and its first partial derivatives. Another criterion is expressed in terms of logarithmic derivations: \(D_1\) and \(D_2\) are splayed if and only if the natural monomorphism NEWLINE\[NEWLINE\frac{\mathrm{Der}_{V,p}}{\mathrm{Der}_{V,p}(-\log (D_1\cup D_2))}\hookrightarrow\frac{\mathrm{Der}_{V,p}}{\mathrm{Der}_{V,p}(-\log D_1)}\oplus \frac{\mathrm{Der}_{V,p}}{\mathrm{Der}_{V,p}(-\log D_2)}NEWLINE\]NEWLINE is an isomorphism. The later formula is related to an expression for Chern-Schwartz-MacPherson class of the divisor complement in terms of logarithmic derivations: NEWLINE\[NEWLINEc_{SM}(V\setminus D)=c(\mathrm{Der}(-\log D))\cap [V]\,.NEWLINE\]NEWLINE The equality does not hold always, but there is a wide class of singularities for which the formula is true. If this formula holds for splayed divisors \(D_1\) and \(D_2\), then NEWLINE\[NEWLINEc(V)\cdot c_{SM}(V\setminus(D_1\cup D_2))=c_{SM}(V\setminus D_1)\cdot c_{SM}(V\setminus D_2).NEWLINE\]NEWLINE It is shown that two divisors on a nonsingular compact surface are splayed if and only if the formula for the Chern-Schwarz-MacPherson classes holds.