Euler characteristics of general linear sections and polynomial Chern classes
Let \(X\) be a locally closed subset of \(\mathbb{P}^n\), and let \(X_r\) be the intersection of \(X\) with \(r\) general hyperplanes. One assembles the Euler characteristics of the general linear sections into a generating polynomial: \[ \chi_X(t):=\sum_{r=0}^n \chi(X_r) \cdot (-t)^r. \] In the paper under review, the author connects \(\chi_X(t)\) with the Chern-Schwartz-MacPherson class of the constructible function \(\mathbf{1}_X\) on \(\mathbb{P}^n\). Consider the polynomial obtained from \(c_{SM}(\mathbf{1}_X) \in A_*(\mathbb{P}^n)\) by replacing the rational equivalence class of the \(r\)-dimensional linear space with \(t^r\): \[ \gamma_X(t):= \sum \gamma_r(X) \cdot t^r \] Theorem 1.1 gives a precise relation between the Chern-Schwartz-MacPherson class and the Euler characteristics of general linear sections: \[ \gamma_X=\mathscr{I}(\chi_X), \qquad \chi_X=\mathscr{I}(\gamma_X), \] where \(\mathscr{I}\) is the involution \[ \mathscr{I}\big(p(t)\big):=\frac{t \cdot p(-t-1)+p(0)}{t+1}. \] The formula can be used, for example, to give a simple proof of a conjecture of Dolgachev that a homogeneous polynomial \(F\) is homaloidal if and only if \(F_{\text{red}}\) is, extending the previous proof by \textit{A. Dimca} and \textit{S. Papadima} [Ann. Math. (2) 158, No. 2, 473--507 (2003; Zbl 1068.32019)] to arbitrary algebraically closed field of characteristic zero. The author also describes a simple approach to a theory of `polynomial Chern classes' for varieties endowed with a morphism to a projective space. The polynomial Chern class gives a homomorphism from a certain relative Grothendieck ring of varieties to \(\mathbb{Z}[t]\), which can be constructed using the theory of Chern-Schwartz-MacPherson classes. Remarkably, this `polynomial' version of the Chern class theory, including an analogue of the key covariance property of \(c_{SM}\), can be established using no tools other than basic properties of the Euler characteristic.
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