"Let X be a complex nonsingular variety with globally generated tangent bundle. We prove that the signed Segre-MacPherson (SM) class of a constructible function on X with effective characteristic cycle is effective. This observation has a surprising number of applications to positivity questions in classical situations, unifying previous results in the literature and yielding several new results. We survey a selection of such results in this paper. For example, we prove general effectivity results for SMclasses of subvarieties which admit proper (semi-)small resolutions and for regular or affine embeddings. Among these, we mention the effectivity of (signed) Segre-Milnor classes of complete intersections if X is projective and an alternation property for SM classes of Schubert cells in flag manifolds; the latter result proves and generalizes a variant of a conjecture of Feh'er and Rim'anyi. Among other applications we prove the positivity of Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety and the signed-effectivity of the intersection homology Chern class of the theta divisor of a non-hyperelliptic curve; and we extend the (known) nonnegativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety"--
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Written to honor the enduring influence of William Fulton, these articles present substantial contributions to algebraic geometry.
14. Stability of tangent bundles on smooth toric Picard-rank-2 varieties
and surfaces Milena Hering, Benjamin Nill and Hendrik Süß;
15. Tropical
cohomology with integral coefficients for analytic spaces Philipp Jell;
16.
Schubert polynomials, pipe dreams, equivariant classes, and a co-transition
formula Allen Knutson;
17. Positivity certificates via integral
representations Khazhgali Kozhasov, Mateusz Michaek and Bernd Sturmfels;
18.
On the coproduct in affine Schubert calculus Thomas Lam, Seung Jin Lee and
Mark Shimozono;
19. BostConnes systems and F1-structures in Grothendieck
rings, spectra, and Nori motives Joshua F. Lieber, Yuri I. Manin, and Matilde
Marcolli;
20. Nef cycles on some hyperkähler fourfolds John Christian Ottem;
21. Higher order polar and reciprocal polar loci Ragni Piene;
22.
Characteristic classes of symmetric and skew-symmetric degeneracy loci
Sutipoj Promtapan and Richįrd Rimįnyi;
23. Equivariant cohomology, Schubert
calculus, and edge labeled tableaux Colleen Robichaux, Harshit Yadav and
Alexander Yong;
24. Galois groups of composed Schubert problems Frank
Sottile, Robert Williams and Li Ying;
25. A K-theoretic Fulton class Richard
P. Thomas.
Paolo Aluffi is Professor of Mathematics at Florida State University. He earned a Ph.D. from Brown University with a dissertation on the enumerative geometry of cubic plane curves, under the supervision of William Fulton. His research interests are in algebraic geometry, particularly intersection theory and its application to the theory of singularities and connections with theoretical physics. David Anderson is Associate Professor of Mathematics at The Ohio State University. He earned his Ph.D. from the University of Michigan, under the supervision of William Fulton. His research interests are in combinatorics and algebraic geometry, with a focus on Schubert calculus and its applications. Milena Hering is Reader in the School of Mathematics at the University of Edinburgh. She earned a Ph.D. from the University of Michigan with a thesis on syzygies of toric varieties, under the supervision of William Fulton. Her research interests are in algebraic geometry, in particular toric varieties, Hilbert schemes, and connections to combinatorics and commutative algebra. Mircea Musta is Professor of Mathematics at the University of Michigan, where he has been a colleague of William Fulton for over 15 years. He received his Ph.D. from the University of California, Berkeley under the supervision of David Eisenbud. His work is in algebraic geometry, with a focus on the study of singularities of algebraic varieties. Sam Payne is Professor in the Department of Mathematics at the University of Texas at Austin. He earned his Ph.D. at the University of Michigan, with a thesis on toric vector bundles, under the supervision of William Fulton. His research explores the geometry, topology, and combinatorics of algebraic varieties and their moduli spaces, often through relations to tropical and nonarchimedean analytic geometry.
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