The Chern-Schwartz-MacPherson class of an embeddable scheme

Abstract

There is an explicit formula expressing the Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety (in characteristic $0$) in terms of the Segre class of its jacobian subscheme; this has been known for a number of years. We generalize this formula to arbitrary embeddable schemes: for every subscheme $X$ of a nonsingular variety $V$, we define an associated subscheme $Y$ of a projective bundle over $V$ and provide an explicit formula for the Chern-Schwartz-MacPherson class of $X$ in terms of the Segre class of $Y$. If $X$ is a local complete intersection, a version of the result yields a direct expression for the Milnor class of $X$.

v2: References added, included a section on the relation with the Segre zeta function

References in corpus (4)

• Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells
• Chern-Schwartz-MacPherson classes of degeneracy loci
• On the Milnor classes of local complete intersections
• A relative Segre zeta function

Cited by in corpus (5)

• Machine Learned Calabi-Yau Metrics and Curvature
• Segre Class Computation and Practical Applications
• Segre classes and invariants of singular varieties
• The double point formula with isolated singularities and canonical embeddings
• An approach to Lagrangian specialisation through MacPherson's graph construction