-
Lorentzian polynomials, Segre classes, and adjoint polynomials of
convex polyhedral cones. arXiv:2304.02043. Advances in Mathematics
437 (2024) 109440. doi:10.1016/j.aim.2023.109440
We consider polynomials expressing the cohomology classes of
subvarieties of products of projective spaces, and limits of positive
real multiples of such polynomials. We study the relation between
these covolume polynomials and Lorentzian polynomials. While
these are distinct notions, we prove that, like Lorentzian
polynomials, covolume polynomials have M-convex support and generalize
the notion of log-concave sequences. In fact, we prove that covolume
polynomials are `sectional log-concave', that is, the coefficients of
suitable restrictions of these polynomials form log-concave sequences.
We observe that Chern classes of globally generated bundles give rise
to covolume polynomials, and use this fact to prove that certain
polynomials associated with Segre classes of subschemes of
products of projective spaces are covolume polynomials. We conjecture
that the same polynomials may be Lorentzian after a standard
normalization operation.
Finally, we obtain a combinatorial application of a particular case of
our Segre class result. We prove that the adjoint polynomial
of a convex polyhedral cone contained in the nonnegative orthant, and
sharing a face with it, is a covolume polynomials. This implies that
these adjoint polynomials are M-convex and sectional log-concave, and
in fact dually Lorentzian, that is, Lorentzian after a certain
change of variables.
-
Shadows of characteristic cycles, Verma modules, and positivity of
Chern-Schwartz-MacPherson classes of Schubert cells,
with Leonardo Mihalcea, Jörg Schürmann, Changjian Su;
arXiv:1708.08697.
Duke Math. J. 172 (2023) 3257--3320. doi:10.1215/00127094-2022-0101
Chern-Schwartz-MacPherson (CSM) classes generalize to singular
and/or noncompact varieties the classical total homology Chern class
of the tangent bundle of a smooth compact complex manifold. The
theory of CSM classes has been extended to the equivariant setting
by Ohmoto. We prove that for an arbitrary complex algebraic manifold
X, the homogenized, torus equivariant CSM class of a constructible
function φ is the restriction of the characteristic cycle of
φ via the zero section of the cotangent bundle of X. This
extends to the equivariant setting results of Ginzburg and
Sabbah. We specialize X to be a (generalized) flag manifold
G/B. In this case CSM classes are determined by a Demazure-Lusztig
(DL) operator. We prove a `Hecke orthogonality' of CSM classes,
determined by the DL operator and its Poincar{\'e} adjoint. We
further use the theory of holonomic DX-modules to show
that the characteristic cycle of the Verma module, restricted to the
zero section, gives the CSM class of a Schubert cell. Since the
Verma characteristic cycles naturally identify with the Maulik and
Okounkov's stable envelopes, we establish an equivalence between CSM
classes and stable envelopes; this reproves results of Rim{\'a}nyi
and Varchenko. As an application, we obtain a Segre type formula for
CSM classes. In the non-equivariant case this formula is manifestly
positive, showing that the expansion in the Schubert basis of the
CSM class of a Schubert cell is effective. This proves a previous
conjecture by Aluffi and Mihalcea, and it extends previous
positivity results by J. Huh in the Grassmann manifold case.
Finally, we generalize all of this to partial flag manifolds G/P.
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Motives of melonic graphs,
with Matilde Marcolli, Waleed Qaisar,
arXiv:2007.08565. Ann. Inst. Henri Poincaré - D 10 (2023),
no. 3, pp. 503-554. doi:10.4171/AIHPD/156
We investigate recursive relations for the Grothendieck classes of
the affine graph hypersurface complements of melonic graphs. We
compute these classes explicitly for several families of melonic
graphs, focusing on the case of graphs with valence-$4$ internal
vertices, relevant to CTKT tensor models. The results hint at a
complex and interesting structure, in terms of divisibility
relations or nontrivial relations between classes of graphs in
different families. Using the recursive relations we prove that the
Grothendieck classes of all melonic graphs are positive as
polynomials in the class of the moduli space $\mathcal M_{0,4}$. We
also conjecture that the corresponding polynomials are {\em
log-concave,\/} on the basis of hundreds of explicit computations.
-
Segre classes and invariants of singular varieties.
arXiv:2109.05061.
In `Handbook of Geometry and Topology of Singularities, III',
Springer (2022) 419-492. doi:10.1007/978-3-030-95760-5_6
Segre classes encode essential intersection-theoretic
information concerning vector bundles and embeddings of schemes. In
this paper we survey a range of applications of Segre classes to the
definition and study of invariants of singular spaces. We will focus
on several numerical invariants, on different notions of
characteristic classes for singular varieties, and on classes of Le
cycles. We precede the main discussion with a review of relevant
background notions in algebraic geometry and intersection theory.
-
Positivity of Segre-MacPherson classes, with Leonardo C. Mihalcea,
Jörg Schürmann, Changjian Su, arXiv:1902.00762.
In `Facets of Algebraic Geometry: A Collection in Honor of William
Fulton's 80th Birthday. Volume 1', Cambridge University Press (2022)
1-28. doi:10.1017/9781108877831.001
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 extends and unifies several previous results in the
literature, and yields several new results. For example, we prove that
Behrend's Donaldson-Thomas invariant for a closed subvariety of an
abelian variety is effective; that the intersection homology Chern
class of the theta divisor for a non-hyperellptic curve is
signed-effective; and we prove more general effectivity results for SM
classes 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{\'e}r and Rim{\'a}nyi.
Finally, we extend the (known) non-negativity of the Euler
characteristic of perverse sheaves on a semi-abelian variety to more
general varieties dominating an abelian variety.
-
Intersection theory, characteristic classes, and algebro-geometric Feynman
rules, with Matilde Marcolli. 36 pages.
In `Proceedings of MathemAmplitudes 2019:
Intersection Theory & Feynman Integrals', Proceedings of Science (2022).
doi:10.22323/1.383.0012
We review the basic definitions in Fulton-MacPherson Intersection
Theory and discuss a theory of `characteristic classes' for
arbitrary algebraic varieties, based on this intersection theory. We
also discuss a class of graph invariants motivated by amplitude
computations in quantum field theory. These `abstract Feynman rules'
are obtained by studying suitable invariants of hypersurfaces
defined by the Kirchhoff-Tutte-Symanzik polynomials of graphs. We
review a `motivic' version of these abstract Feynman rules, and
describe a counterpart obtained by intersection-theoretic
techniques.
-
Newton-Okounkov bodies and Segre classes,
arXiv:1809.07344. Amer. J. Math. 143 (2021) 1505-1526.
doi:10.1353/ajm.2021.0038
Given a homogeneous ideal in a polynomial ring over C, we adapt the
construction of Newton-Okounkov bodies to obtain a convex subset of
Euclidean space such that a suitable integral over this set computes
the Segre zeta function of the ideal. That is, we extract the
numerical information of the Segre class of a subscheme of projective
space from an associated (unbounded) Newton-Okounkov convex set. The
result generalizes to arbitrary subschemes of projective space the
numerical form of a previously known result for monomial schemes.
-
Pfaffian integrals and invariants of singular varieties,
with Mark Goresky, arXiv:1901.06312. A panorama of singularities,
1-11. Contemp. Math., 742 (2020). doi:10.1090/conm/742/14935
Integrals of the Pfaffian form over the nonsingular part of a
projective variety compute information closely related to the
Mather-Chern class of the variety and to other invariants such as the
local Euler obstruction along strata of its singular locus and, in the
hypersurface case, Milnor numbers. We obtain simple proofs of these
formulas, recovering along the way several classically known
results.
-
The Chern-Schwartz-MacPherson class of an embeddable scheme,
arXiv:1805.11116. Forum of Mathematics, Sigma (2019), 7, 28 pages.
doi:10.1017/fms.2019.25
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.
-
The Euclidean Distance degree of smooth complex projective varieties,
with Corey Harris; arXiv:1708.00024. Algebra & Number Theory 12:8
(2018) 2005-2032. doi:10.2140/ant.2018.12.2005
We obtain several formulas for the Euclidean distance degree (ED
degree) of an arbitrary nonsingular variety in projective space: in
terms of Chern and Segre classes, Milnor classes,
Chern-Schwartz-MacPherson classes, and an extremely simple formula
equating the Euclidean distance degree of X with the Euler
characteristic of an open subset of X.
-
Projective duality and a Chern-Mather involution,
arXiv:1601.05427.
Trans. Amer. Math. Soc. 370 (2018), no. 3, 1803--1822.
doi:10.1090/tran/7042
We observe that linear relations among Chern-Mather classes of
projective varieties are preserved by projective duality. We deduce
the existence of an explicit involution on a part of the Chow group of
projective space, encoding the effect of duality on Chern-Mather
classes. Applications include Plücker formulae, constraints on
self-dual varieties, generalizations to singular varieties of
classical formulas for the degree of the dual and the dual defect,
formulas for the Euclidean distance degree, and computations of
Chern-Mather classes and local Euler obstructions for cones.
-
The Segre zeta function of an ideal,
arXiv:1606.03098. Advances in Mathematics 320 (2017) 1201-1226.
doi:10.1016/j.aim.2017.09.023
We define a power series associated with a homogeneous ideal in a
polynomial ring, encoding information on the Segre classes defined by
extensions of the ideal in projective spaces of arbitrarily high
dimension. We prove that this power series is rational, with poles
corresponding to generators of the ideal, and with numerator of
bounded degree and with nonnegative coefficients. We also prove that
this `Segre zeta function' only depends on the integral closure of the
ideal.
The results follow from good functoriality properties of the `shadows'
of rational equivalence classes of projective bundles. More precise
results can be given if all homogeneous generators have the same
degree, and for monomial ideals.
In certain cases, the general description of the Segre zeta function
given here leads to substantial improvements in the speed of
algorithms for the computation of Segre classes. We also compute the
projective ranks of a nonsingular variety in terms of the
corresponding zeta function, and we discuss the Segre zeta function of
a local complete intersection of low codimension in projective space.
-
Tensored Segre classes, arXiv:1605.09393.
Journal of Pure and Applied Algebra 221 (2017) 1366--1382.
doi:10.1016/j.jpaa.2016.09.016
We study a class obtained from the Segre class s(Z,Y)
of an embedding of schemes by incorporating the datum of a line
bundle on Z. This class satisfies basic properties analogous to
the ordinary Segre class, but leads to remarkably simple formulas in
standard intersection-theoretic situations such as excess or
residual intersections. We prove a formula for the behavior of this
class under linear joins, and use this formula to prove that a `Segre
zeta function' associated with ideals generated by forms of the same
degree is a rational function.
-
How many hypersurfaces does it take to cut out a Segre class?,
arXiv:1605.00012, Journal of Algebra 471 (2017) 480-491.
doi:10.1016/j.jalgebra.2016.10.003
We prove an identity of Segre classes for zero-schemes of compatible
sections of two vector bundles. Applications include bounds on the
number of equations needed to cut out a scheme with the same Segre
class as a given subscheme of (for example) a projective variety, and a
`Segre-Bertini' theorem controlling the behavior of Segre classes of
singularity subschemes of hypersurfaces under general hyperplane
sections.
These results interpolate between an observation of Samuel concerning
multiplicities along components of a subscheme and facts concerning
the integral closure of corresponding ideals. The Segre-Bertini theorem
has applications to characteristic classes of singular varieties.
The main results are motivated by the problem of computing Segre classes
explicitly and applications of Segre classes to enumerative geometry.
-
Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds,
with Leonardo Mihalcea,
arXiv:1508.01535. Compositio Math.~152 (2016), 2603--2625.
doi:10.1112/S0010437X16007685.
We obtain an algorithm describing the Chern-Schwartz-MacPherson
(CSM) classes of Schubert cells in generalized flag manifolds $G/B$.
In analogy to how the ordinary divided difference operators act on
Schubert classes, each CSM class of a Schubert class of a Schubert
cell is obtained by applying certain Demazure-Lusztig type operators to the
CSM class of a cell of dimension one less. By functoriality, we deduce
algorithmic expressions for CSM classes of Schubert cells in any flag
manifold $G/P$. We conjecture that the CSM classes of Schubert
cells are an effective combination of (homology) Schubert classes,
and prove that this is the case in several classes of examples.
-
Segre classes as integrals over polytopes, arXiv:1307.0830.
J. Eur. Math. Soc. 18 (2016) 2849-2863.
doi:10.4171/JEMS/655
We express the Segre class of a monomial scheme---or, more
generally, a scheme monomially supported on a set of divisors cutting
out complete intersections---in terms of an integral computed over an
associated body in euclidean space. The formula is in the spirit of
the classical Bernstein-Kouchnirenko theorem computing intersection
numbers of equivariant divisors in a torus in terms of mixed volumes,
but deals with the more refined intersection-theoretic invariants
given by Segre classes, and holds in the less restrictive context of
`r.c.-monomial schemes'.
-
Multidegrees of monomial rational maps. arXiv:1308.4152.
Publ. RIMS Kyoto Univ. 51 (2015), 635-654. doi:10.4171/PRIMS/167
We prove a formula for the multidegrees of a rational map defined
by generalized monomials on a projective variety, in terms of
integrals over an associated Newton region. This formula leads to an
expression of the multidegrees as volumes of related polytopes, in the
spirit of the classical Bernstein-Kouchnirenko theorem, but extending
the scope of these formulas to more general monomial maps. We also
determine a condition under which the multidegrees may be computed in
terms of the characteristic polynomial of an associated matrix.
-
Chern classes of splayed intersections, with E. Faber. arXiv:1406.1182.
Canad. J. Math. 67(2015), 1201-1218,
doi: 10.4153/CJM-2015-010-7
We generalize the Chern class relation for the transversal
intersection of two nonsingular varieties to a relation for possibly
singular varieties, under a 'splayedness' assumption. The relation is
shown to hold for both the Chern-Schwartz-MacPherson class and the
Chern-Fulton class. The main tool is a formula for Segre classes of
splayed subschemes. We also discuss the Chern class relation under the
assumption that one of the varieties is a general very ample
divisor.
-
Degrees of projections of rank loci. arXiv:1408.1702.
Experimental Mathematics, 24 (2015) 469--488.
DOI: 10.1080/10586458.2015.1028689.
We provide formulas for the degrees of the projections of the locus
of square matrices with given rank from linear spaces spanned by a
choice of matrix entries. The motivation for these computations stem
from applications to `matrix rigidity'; we also view them as an
excellent source of examples to test methods in intersection theory,
particularly computations of Segre classes. Our results are generally
expressed in terms of intersection numbers in Grassmannians, which can
be computed explicitly in many cases. We observe that, surprisingly
(to us), these degrees appear to match the numbers of Kekul\'e
structures of certain `benzenoid hydrocarbons', and arise in many
other contexts with no apparent direct connection to the enumerative
geometry of rank conditions.
-
Log canonical threshold and Segre classes of monomial schemes.
arXiv:1308.4153. Manu. Math. (2015) 146: 1--6. DOI:10.1007/s00229-014-0686-6
We express the Segre class of a monomial scheme in projective space
in terms of log canonical thresholds of associated ideals. Explicit
instances of the relation amount to identities involving the classical
polygamma functions.
-
Generalized Euler characteristics, graph hypersurfaces,
and Feynman periods. In "Geometric, Algebraic and Topological Methods
for Quantum Field Theory", proceedings of the Summer
school in Villa de Leyva, 2011. Cardona, Meira-Jimenez, Ocampo, Paycha,
Reyes-Lega, editors. World Scientific, 2014.
We give a very informal presentation of background on the
Grothendieck group of varieties and on characteristic classes, both
viewed as generalizations of the ordinary topological Euler
characteristic. We then review some recent work using these tools to
study `graph hypersurfaces'---a topic motivated by the
algebro-geometric interpretation of Feynman amplitudes as periods of
complements of these hypersurfaces. These notes follow closely, both
in content and style, my lectures at the Summer school in Villa de
Leyva, July 5-8, 2011.
-
Splayed divisors and their Chern classes, with Eleonore Faber,
arXiv:1207.4202. J. London Math. Soc. (2013) 88 (2): 563-579.
DOI:10.1112/jlms/jdt032
We obtain several new characterizations of splayedness for
divisors: a Leibniz property for ideals of singularity subschemes, the
vanishing of a `splayedness' module, and the requirements that certain
natural morphisms of modules and sheaves of logarithmic derivations
and logarithmic differentials be isomorphisms. We also consider the
effect of splayedness on the Chern classes of sheaves of differential
forms with logarithmic poles along splayed divisors, as well as on the
Chern-Schwartz-MacPherson classes of the complements of these
divisors. A postulated relation between these different notions of
Chern class leads to a conjectural identity for
Chern-Schwartz-MacPherson classes of splayed divisors and
subvarieties, which we are able to verify in several template
situations.
-
Segre classes of monomial schemes, arXiv:1303.0010.
Electron. Res. Announc. Math. Sci., 20 (2013), 55-70.
DOI:10.3934/era.2013.20.55
We propose an explicit formula for the Segre classes of monomial
subschemes of nonsingular varieties, such as schemes defined by
monomial ideals in projective space. The Segre class is expressed as
a formal integral on a region bounded by the corresponding Newton
polyhedron. We prove this formula for monomial ideals in two variables
and verify it for some families of examples in any number of
variables.
-
Euler characteristics of general linear sections and polynomial
Chern classes. Rend. Circ. Mat. Palermo. (special issue) 62 (2013) 3-26.
DOI:10.1007/s12215-013-0106-x
We obtain a precise relation between the Chern-Schwartz-MacPherson
class of a subvariety of projective space and the Euler
characteristics of its general linear sections. In the case of a
hypersurface, this leads to simple proofs of formulas of
Dimca-Papadima and Huh for the degrees of the polar map of a
homogeneous polynomial, extending these formula to any algebraically
closed field of characteristic~$0$, and proving a conjecture of
Dolgachev on `homaloidal' polynomials in the same context. We
generalize these formulas to subschemes of higher codimension in
projective space.
We also describe a simple approach to a theory of `polynomial
Chern classes' for varieties endowed with a morphism to projective
space, recovering properties analogous to the Deligne-Grothendieck
axioms from basic properties of the Euler characteristic. We prove
that the polynomial Chern class defines homomorphisms from suitable
relative Grothendieck rings of varieties to Zbb[t].
-
A motivic approach to phase transitions in Potts models, with
Matilde Marcolli. Journal of Geometry and Physics 63 (2013), 6-31.
DOI: 10.1016/j.geomphys.2012.09.003
We describe an approach to the study of phase transitions in Potts
models based on an estimate of the complexity of the locus of real
zeros of the partition function, computed in terms of the classes in
the Grothendieck ring of the affine algebraic varieties defined by the
vanishing of the multivariate Tutte polynomial. We give completely
explicit calculations for the examples of the chains of linked
polygons and of the graphs obtained by replacing the polygons with
their dual graphs. These are based on a deletion-contraction formula
for the Grothendieck classes and on generating functions for splitting
and doubling edges.
-
Chern classes of graph hypersurfaces and deletion-contraction relations,
Moscow Math J. 12 (2012) 671-700.
We study the behavior of the Chern classes of graph hypersurfaces
under the operation of deletion-contraction of an edge of the
corresponding graph. We obtain an explicit formula when the edge
satisfies two technical conditions, and prove that both these
conditions hold when the edge is multiple in the graph. This leads to
recursions for the Chern classes of graph hypersurfaces for graphs
obtained by adding parallel edges to a given (regular) edge.
Analogous results for the case of Grothendieck classes of graph
hypersurfaces were obtained in previous work. Both Grothendieck
classes and Chern classes were used to define `algebro-geometric'
Feynman rules. The results in this paper provide further evidence
that the polynomial Feynman rule defined in terms of the
Chern-Schwartz-MacPherson class of a graph hypersurface reflects
closely the combinatorics of the corresponding graph.
The key to the proof of the main result is a more general formula for the
Chern-Schwartz-MacPherson class of a transversal intersection
(see section 3), which may be of independent interest.
We also describe a more geometric approach, using the apparatus of
`Verdier specialization'.
-
Chern classes of free hypersurface arrangements,
J. Sing. 5 (2012) 19-32.
DOI: 10.5427/jsing.2012.5b
The Chern class of the sheaf of logarithmic derivations along a
simple normal crossing divisor equals the Chern-Schwartz-MacPherson
class of the complement of the divisor. We extend this equality to
more general divisors, which are locally analytically isomorphic to
free hyperplane arrangements.
-
Grothendieck classes and Chern classes of hyperplane arrangements,
Int. Math. Res. Notices (2013) 1873-1900.
DOI: 10.1093/imrn/rns100
We show that the characteristic polynomial of a hyperplane
arrangement can be recovered from the class in the Grothendieck group
of varieties of the complement of the arrangement. This gives a quick
proof of a theorem of Orlik and Solomon relating the characteristic
polynomial with the ranks of the cohomology of the complement of the
arrangement.
We also show that the characteristic polynomial can be computed
from the total Chern class of the complement of the arrangement; this
has also been observed by Huh. In the case of free arrangements, we
prove that this Chern class agrees with the Chern class of the dual of
a bundle of differential forms with logarithmic poles along the
hyperplanes in the arrangement; this follows from work of Mustata
and Schenck. We conjecture that this relation holds for all free
divisors.
We give an explicit relation between the characteristic polynomial
of an arrangement and the Segre class of its singularity (`Jacobian')
subscheme. This gives a variant of a recent result of Wakefield and
Yoshinaga, and shows that the Segre class of the singularity subscheme
of an arrangement together with the degree of the arrangement
determine the ranks of the cohomology of its complement.
We also discuss the positivity of the Chern classes of hyperplane
arrangements: we give a combinatorial interpretation of this
phenomenon, and we discuss the cases of generic and free arrangements.
-
Verdier specialization via weak factorization.
Arkiv foer Matematik 51 (2013) 1-28. DOI:10.1007/s11512-011-0164-2
Let X in V be a closed embedding, with V - X nonsingular. We define
a constructible function on X, agreeing with Verdier's specialization
of the constant function 1 when X is the zero-locus of a function on
V. Our definition is given in terms of an embedded resolution of X;
the independence on the choice of resolution is obtained as a
consequence of the weak factorization theorem of Abramovich et al.
The main property of the specialization function is a compatibility
with the specialization of the Chern class of the complement V-X. With
the definition adopted here, this is an easy consequence of standard
intersection theory. It recovers Verdier's result when X is the
zero-locus of a function on V.
Our definition has a straightforward counterpart in a motivic
group. The specialization function and the corresponding Chern class
and motivic aspect all have natural `monodromy' decompositions, for
for any X in V as above.
The definition also yields an expression for Kai Behrend's
constructible function when applied to (the singularity subscheme of)
the zero-locus of a function on V.
-
Feynman motives and deletion-contraction relations
(with Matilde Marcolli).
In Topology of Algebraic Varieties and Singularities, Cont. Math.,
vol. 538, Amer. Math. Soc., Providence, RI, 2011, pp. 21-64.
doi:10.1090/conm/538
We prove a deletion-contraction formula for motivic Feynman rules given by
the classes of the affine graph hypersurface complement in the Grothendieck
ring of varieties.
We derive explicit recursions and generating series for these motivic
Feynman rules under the operation of multiplying edges in a
graph and we compare it with similar formulae for the Tutte
polynomial of graphs, both being specializations of the same universal
recursive relation. We obtain similar recursions for graphs that are
chains of polygons and for graphs obtained by replacing an edge by
a chain of triangles. We show that the deletion-contraction relation
can be lifted to the level of the category of mixed motives in the form
of a distinguished triangle, similarly to what happens in categorifications
of graph invariants.
-
Algebro-Geometric Feynman rules
(with Matilde Marcolli),
Int. Jour. of Geom. Methods in Modern Phys. 8 (2011) 203-237.
DOI 10.1142/S0219887811005099
We give a general procedure to construct algebro-geometric Feynman
rules, that is, characters of the Connes--Kreimer Hopf algebra of
Feynman graphs that factor through a Grothendieck ring of immersed
conical varieties, via the class of the complement of the affine graph
hypersurface. In particular, this maps to the usual Grothendieck ring
of varieties, defining motivic Feynman rules. We also construct an
algebro-geometric Feynman rule with values in a polynomial ring, which
does not factor through the usual Grothendieck ring, and which is
defined in terms of characteristic classes of singular varieties. This
invariant recovers, as a special value, the Euler characteristic of
the projective graph hypersurface complement. The main result
underlying the construction of this invariant is a formula for the
characteristic classes of the join of two projective varieties. We
discuss the BPHZ renormalization procedure in this algebro-geometric
context and some motivic zeta functions arising from the partition
functions associated to motivic Feynman rules.
-
Graph hypersurfaces and a dichotomy in the Grothendieck ring
(with Matilde Marcolli).
Lett Math Phys (2011) 95:223-232.
DOI 10.1007/s11005-011-0461-5.
The subring of the Grothendieck ring of varieties generated by the
graph hypersurfaces of quantum field theory maps to the monoid ring of
stable birational equivalence classes of varieties. We show that the
image of this map is the copy of Z generated by the class of a
point.
Thus, the span of the graph hypersurfaces in the Grothendieck ring
is nearly killed by setting the Lefschetz motive L to zero,
while it is known that graph hypersurfaces generate the Grothendieck
ring over a localization of Z[L] in which L
becomes invertible. In particular, this shows that the graph
hypersurfaces do
not generate the Grothendieck ring prior to localization.
The same result yields some information on the mixed Hodge
structures of graph hypersurfaces, in the form of a constraint on the
terms in their Deligne-Hodge polynomials.
-
Parametric Feynman integrals and determinant hypersurfaces
(with Matilde Marcolli).
Adv. Theor. Math. Phys. 14 (2010) 911-963.
DOI: 10.4310/ATMP.2010.v14.n3.a5
The purpose of this paper is to show that, under certain
combinatorial conditions on the graph, parametric Feynman integrals
can be realized as periods on the complement of the determinant
hypersurface in an affine space depending on the number of loops of
the Feynman graph. The question of whether the Feynman integrals are
periods of mixed Tate motives can then be reformulated (modulo
divergences) as a question on a relative cohomology being a
realization of a mixed Tate motive. This is the cohomology of the pair
of the determinant hypersurface complement and a normal crossings
divisor depending only on the number of loops and the genus of the
graph. We show explicitly that this relative cohomology is a
realization of a mixed Tate motive in the case of three loops and we
give alternative formulations of the main question in the general
case, by describing the locus of intersection of the divisor with the
determinant hypersurface complement in terms of intersections of
unions of Schubert cells in flag varieties. We also discuss different
methods of regularization aimed at removing the divergences of the
Feynman integral.
-
New orientifold weak coupling limits in F-theory
(with Mboyo Esole).
J. High Energy Phys. (2010) no. 2, 1-53.
DOI: 10.1007/JHEP02(2010)020
We present new explicit constructions of weak coupling limits of
F-theory generalizing Sen's construction to elliptic fibrations which
are not necessary given in a Weierstrass form. These new constructions
allow for an elegant derivation of several brane configurations that
do not occur within the original framework of Sen's limit, or which
would require complicated geometric tuning or break supersymmetry. Our
approach is streamlined by first deriving a simple geometric
interpretation of Sen's weak coupling limit. This leads to a natural
way of organizing all such limits in terms of transitions from
semistable to unstable singular fibers. These constructions provide a
new playground for model builders as they enlarge the number of
supersymmetric configurations that can be constructed in F-theory. We
present several explicit examples for E8, E7 and E6 elliptic
fibrations.
-
Chern classes of blow-ups.
Math Proc. of the Cambridge Phil. Soc. (2010) 148: 227-242.
doi:10.1017/S0305004109990247.
We extend the classical formula of Porteous for blowing-up Chern classes
to the case of blow-ups of possibly singular varieties along regularly
embedded centers.
The proof of this generalization is perhaps conceptually simpler than the
standard argument for the nonsingular case, involving Riemann-Roch
without denominators. The new approach
relies on the explicit computation of an ideal, and a mild generalization
of a well-known formula for the normal bundle of a proper transform.
We also discuss alternative, very short proofs of the standard
formula in some cases: an approach relying on the theory of
Chern-Schwartz-MacPherson classes (working in characteristic 0), and
an argument reducing the formula to a straightforward computation of
Chern classes for sheaves of differential 1-forms with logarithmic
poles (when the center of the blow-up is a complete intersection).
-
Limits of PGL(3)-translates of plane curves, II
(with Carel Faber),
Journal of Pure and Applied Algebra 214 (2010), 548-564.
doi:10.1016/j.jpaa.2009.06.012.
Every complex plane curve C determines a subscheme S
of the P8 of 3x3 matrices, whose projective
normal cone (PNC) captures subtle invariants of C.
In a previous paper (FSU07-15) we obtain a set-theoretic
description of the PNC and thereby we determine all possible limits of
families of plane curves whose general element is isomorphic to
C. The main result of this article is the determination of the
PNC as a cycle; this is an essential ingredient in our
computation in Linear orbits of arbitrary plane curves,
Michigan Math J., 48 (2000) 1-37, of the degree of the
PGL(3)-orbit closure of an arbitrary plane curve, an invariant of
natural enumerative significance.
-
Limits of PGL(3)-translates of plane curves, I
(with Carel Faber),
Journal of Pure and Applied Algebra 214 (2010), 526-547.
doi:10.1016/j.jpaa.2009.06.010.
We classify all possible limits of families of translates of a fixed,
arbitrary complex plane curve. We do this by giving a set-theoretic
description of the projective normal cone (PNC) of a subscheme,
determined by the curve, of the P8 of 3x3 matrices.
In a sequel to this paper we determine the multiplicities of the
components of the PNC. The knowledge of the PNC as a cycle is
essential in our computation of the
degree of the PGL(3)-orbit closure of an arbitrary plane curve,
performed in Linear orbits of arbitrary plane curves,
Michigan Math J., 48 (2000) 1-37.
-
Feynman motives of banana graphs
(with Matilde Marcolli), Comm. in Number Theory and Physics (2009)
1-57.
We consider the infinite family of Feynman graphs known as the
``banana graphs'' and compute explicitly the classes of the
corresponding graph hypersurfaces in the Grothendieck ring of
varieties as well as their Chern--Schwartz--MacPherson classes, using
the classical Cremona transformation and the dual graph, and a blowup
formula for characteristic classes. We outline the interesting
similarities between these operations and we give formulae for cones
obtained by simple operations on graphs. We formulate a positivity
conjecture for characteristic classes of graph hypersurfaces and
discuss briefly the effect of passing to noncommutative spacetime.
-
Chern class identities from tadpole matching in type IIB and
F-theory
(with Mboyo Esole),
J. High Energy Phys. (2009) no. 3, 032, 29 pp.
In light of Sen's weak coupling limit of F-theory as a type IIB
orientifold, the compatibility of the tadpole conditions leads to a
non-trivial identity relating the Euler characteristics of an
elliptically fibered Calabi-Yau fourfold and of certain related
surfaces. We present the physical argument leading to the identity,
and a mathematical derivation of a Chern class identity which confirms
it, after taking into account singularities of the relevant loci. This
identity of Chern classes holds in arbitrary dimension, and for
varieties that are not necessarily Calabi-Yau. Singularities are
essential in both the physics and the mathematics arguments: the
tadpole relation may be interpreted as an identity involving stringy
invariants of a singular hypersurface, and corrections for the
presence of pinch-points. The mathematical discussion is streamlined
by the use of Chern-Schwartz-MacPherson classes of singular
varieties. We also show how the main identity may be obtained by
applying `Verdier specialization' to suitable constructible functions.
-
Chern classes of Schubert cells and varieties, with Leonardo
Mihalcea, J. Algebraic Geom. 18 (2009), 37-61
We give explicit formulas for the Chern-Schwartz-MacPherson classes of all Schubert varieties in the Grassmannian of $d$-planes in a vector space, and conjecture that these classes are effective. We prove this is the case for (very) small values of $d$.
-
Une nouvelle preuve de la concordance des classes définies
par M.-H. Schwartz et par R. MacPherson,
(French and English versions)
with J.-P. Brasselet, Bull. Soc. math. France 136 (2008) 159-166
We give a short proof of the fact that the Chern classes for
singular varieties defined by Marie-Hélène Schwartz by
means of "radial frames" agree with the functorial notion defined by
Robert MacPherson.
-
Celestial integration, stringy invariants, and Chern-Schwartz-MacPherson
classes, Real and complex singularities, 1-13, Trends Math.,
Birkhäuser, Basel, 2007.
We introduce a formal integral on the system of varieties mapping
properly and birationally to a given one, with value in an associated
Chow group. Applications include comparisons of Chern numbers of
birational varieties, new birational invariants, `stringy' Chern
classes, and a `celestial' zeta function specializing to the
topological zeta function.
In its simplest manifestation, the integral gives a new expression for
Chern-Schwartz-MacPherson classes of possibly singular varieties,
placing them into a context in which a `change of variable' formula
holds.
The formalism has points of contact with motivic integration.
-
Limits of Chow groups, and a new construction of Chern-Schwartz-MacPherson
classes, Pure Appl. Math. Q. 2 (2006), no. 4, 915-941.
doi:10.4310/PAMQ.2006.v2.n4.a2
We define an `enriched' notion of Chow groups for algebraic
varieties, agreeing with the conventional notion for complete
varieties, but enjoying a functorial push-forward for arbitrary
maps. This tool allows us to glue intersection-theoretic information
across elements of a stratification of a variety; we illustrate this
operation by giving a direct construction of
Chern-Schwartz-MacPherson classes of singular varieties,
providing a new proof of an old (and long since settled) conjecture of
Deligne and Grothendieck.
-
Classes de Chern des variétés singulières,
revisitées,
(French and English versions)
C. R. Math. Acad. Sci. Paris 342 (2006), no. 6, 405-410.
doi:10.1016/j.crma.2006.01.002
We introduce the notion of {\em proChow group\/} of varieties,
agreeing with the notion of Chow group for complete varieties and
covariantly functorial with respect to {\em arbitrary\/} morphisms. We
construct a natural transformation from the functor of constructible
functions to the proChow functor, extending MacPherson's natural
transformation. We illustrate the result by providing very short
proofs of (a generalization of) two well-known facts on
Chern-Schwartz-MacPherson classes.
-
Modification systems and integration in their Chow groups,
Selecta Mathematica 11 (2005) 155-202.
We introduce a notion of integration on the category of proper
birational maps to a given variety $X$, with value in an associated
Chow group. Applications include new birational invariants; comparison
results for Chern classes and numbers of nonsingular birational
varieties; `stringy' Chern classes of singular varieties; and
a zeta function specializing to the topological zeta function.
In its simplest manifestation, the integral gives a new expression for
Chern-Schwartz-MacPherson classes of possibly singular varieties,
placing them into a context in which a `change-of-variable' formula
holds.
-
Characteristic classes of singular varieties,
in Topics in cohomological studies of algebraic
varieties, Birkhäuser (2005) 1-32. Notes for a cycle of five
lectures given at the Banach Center, Warsaw, April 23-27 2002.
doi:10.1007/b137662
These five lectures aim to explain an algebro-geometric
approach to the study of different notions of Chern classes for
singular varieties, with emphasis on results leading to concrete
computations.
The notes are organized so that every page deals with essentially one
topic (a device which I am borrowing from Marvin Minsky's The
Society of Mind). Every one of the five lectures consists of five
pages.
My main goal in the lectures was not to summarize the history or to
give a complete, detailed treatment of the subject; five lectures
would not suffice for this purpose, and I doubt I would be able to
accomplish it in any amount of time anyway. My goal was simply to
provide enough information so that interested listeners could
start working out examples on their own. As these notes are little
more than a transcript of my lectures, they are bound to suffer from
the same limitations. In particular, I am certainly not quoting here
all the sources that should be quoted; I offer my apologies to any
author that may feel his or her contribution has been neglected.
The lectures were given in the mini-school with the same title
organized by Professors Pragacz and Weber at the Banach Center.
Jöorg Schürmann gave a parallel cycle of lectures at the same
mini-school, on the same topic but from a rather different viewpoint.
I believe everybody involved found the counterpoint provided by the
accostment of the two approaches very refreshing. I warmly thank Piotr
Pragacz and Andrzej Weber for giving us the opportunity to present
this beautiful subject.
-
Shadows of blow-up algebras, Tohoku Math. J. 56 (2004) 593-619.
We study different notions of blow-up of a scheme X along a
subscheme Y, depending on the datum of an embedding of X into an
ambient scheme. The two extremes in this theory are the ordinary
blow-up, BlYX, corresponding to the identity X -> X,
and the `quasi-symmetric blow-up', BlYX, corresponding to an
embedding X -> M into a nonsingular variety M. We prove that this latter
blow-up is intrinsic of Y and X, and is universal with respect to the
requirement of being embedded as a subscheme of the ordinary blow-up of
some ambient space along Y.
We consider these notions in the context of the theory of characteristic
classes of singular varieties. We prove that if X a hypersurface in a
nonsingular variety and Y is its `singularity subscheme', these two
extremes embody respectively the conormal and characteristic
cycles of X. Consequently, the first carries the essential information
computing Chern-Mather classes, and the second is likewise a carrier for
Chern-Schwartz-MacPherson classes. In our approach, these classes are
obtained from Segre class-like invariants, in precisely the same way as
other intrinsic characteristic classes such as those proposed by William
Fulton, and by W. Fulton and Kent Johnson.
We also identify a condition on the singularities of a hypersurface
under which the quasi-symmetric blow-up is simply the linear fiber space
associated with a coherent sheaf.
-
Chern classes of birational varieties,
IMRN 63 (2004) 3367-3377.
DOI:10.1155/S1073792804140439
Let $φ: V\dashrightarrow W$ be a birational map between
smooth algebraic varieties which does not change the canonical class
(in the sense of Batyrev). We prove that the total homology Chern
classes of $V$ and $W$ are push-forwards of the same class from a
resolution of indeterminacies of $φ$.
For example, it follows that the push-forward of the total Chern class
of a crepant resolution of a singular variety is independent of the
resolution.
-
Computing characteristic classes of projective schemes,
Journal of Symbolic Computation 35 (2003) 3-19.
Related code and
examples
We discuss an algorithm computing the push-forward to projective space
of several classes associated to a (possibly singular, reducible,
nonreduced) projective scheme. For example, the algorithm yields the
topological Euler characteristic of the support of a projective scheme
$S$, given the homogeneous ideal of $S$. The algorithm has been
implemented in Macaulay2, and it is available here.
-
Inclusion-exclusion and Segre classes, II;
Topics in algebraic and noncommutative geometry (Luminy/Annapolis, MD,
2001), Contemp. Math., 324 (2003) 51-61.
Considerations based on the known relation between different
characteristic classes for singular hypersufaces suggest that a form
of the `inclusion-exclusion' principle may hold for Segre classes. We
formulate and prove such a principle for a notion closely related to
Segre classes. This is used to provide a simple computation of the
classes introduced in our previous work, in certain special (but
representative) cases.
-
Inclusion-exclusion and Segre classes, Comm. Algebra 31 (2003) 3619-3630.
We propose a variation of the notion of Segre class, by forcing a
naive `inclusion-exclusion' principle to hold. The resulting class is
computationally tractable, and is closely related to
Chern-Schwartz-MacPherson classes. We deduce several general
properties of the new class from this relation, and obtain an
expression for the Milnor class of an arbitrary scheme in terms of
this class.
-
Interpolation of characteristic classes of singular hypersurfaces,
with Jean-Paul Brasselet, Adv. in Math 180 (2003) 692-704.
We show that the Chern-Schwartz-MacPherson class of a hypersurface
X in a nonsingular variety M `interpolates' between two
other notions of characteristic classes for singular varieties,
provided that the singular locus of X is smooth and that certain
numerical invariants of X are constant along this locus. This
allows us to define a lift of the Chern-Schwartz-MacPherson class of
such `nice' hypersurfaces to intersection homology. As another
application, the interpolation result leads to an explicit formula for
the Chern-Schwartz-MacPherson class of X in terms of its polar
classes.
-
Weighted Chern-Mather classes and Milnor classes of hypersurfaces,
Singularities and Arrangements, Sapporo-Tokyo 1998,
Advanced
Studies in Pure Mathematics 29 (2000) 1-20.
We introduce a class extending the notion of Chern-Mather class to
possibly nonreduced schemes, and use it to express the difference
between Schwartz- MacPherson's Chern class and the class of the
virtual tangent bundle of a singular hypersurface of a nonsingular
variety. Applications include constraints on the possible
singularities of a hypersurface and on contacts of nonsingular
hypersurfaces, and multiplicity computations.
-
Linear orbits of arbitrary plane curves, with Carel Faber,
Michigan Math. Jour., 48 (2000), 1-37.
The `linear orbit' of a plane curve of degree d is its
orbit in \P^{d(d+3)/2} under the natural action of \PGL(3). In
this paper we obtain an algorithm computing the degree of the closure
of the linear orbit of an arbitrary plane curve, and give explicit
formulas for plane curves with irreducible singularities. The main
tool is an intersection-theoretic study of the projective normal cone
of a scheme determined by the curve in the projective space \P^8 of
3x3 matrices; this expresses the degree of the orbit closure
in terms of the degrees of suitable loci related to the limits of the
curve. These limits, and the degrees of the corresponding loci, have
been established in previous work.
-
Plane curves with small linear orbits II, with Carel Faber,
International Journal of Mathematics, 11 (2000), 591-608.
The `linear orbit' of a plane curve of degree d is its orbit in
P^{d(d+3)/2} under the natural action of PGL(3). We classify curves
with positive dimensional stabilizer, and we compute the degree of the
closure of the linear orbits of curves supported on unions of lines.
Together with the results of [3], this encompasses the enumerative
geometry of all plane curves with small linear orbit. This information
will serve elsewhere as an ingredient in the computation of the degree
of the orbit closure of an arbitrary plane curve.
-
Plane curves with small linear orbits I, with Carel Faber,
Annales de l'institut Fourier, 50 (2000), 151-196.
The `linear orbit' of a plane curve of degree d is its
orbit in P^{d(d+3)/2} under the natural action of PGL(3). In
this paper we compute the degree of the closure of the linear orbits
of most curves with positive dimensional stabilizers. Our tool is a
nonsingular variety dominating the orbit closure, which we construct
by a blow-up sequence mirroring the sequence yielding an embedded
resolution of the curve.
The results given here will serve as an ingredient in the computation
of the analogous information for arbitrary plane curves. Linear
orbits of smooth plane curves are studied in [A-F1].
-
Differential forms with logarithmic poles and
Chern-Schwartz-MacPherson classes of singular varieties,
Comptes Rendus de l'Acad. des Sc., S. I, 329 (1999), 619-624.
We express the Chern-Schwartz-MacPherson class of a possibly
singular variety in terms of the total Chern class of a bundle of
forms with logarithmic poles. As an application, we obtain a formula
for the Chern-Schwartz-MacPherson class of a hypersurface of a
nonsingular variety, in terms of the Chern-Mather class of a suitable
sheaf.
-
Chern classes for singular hypersurfaces,
Trans. Amer. Math. Soc. 351 (1999), 3989-4026.
We prove a formula relating Chern-Schwartz-MacPherson's
class of a hypersurface in a nonsingular variety to other
definitions of homology Chern classes of singular varieties, such as
Mather's Chern class and a class introduced by W. Fulton.
-
Characteristic classes of discriminants and enumerative geometry,
Comm. in Algebra 26(10), 3165-3193 (1998).
We compute the Euler obstruction and Mather's Chern class of
the discriminant hypersurface of a very ample linear system on a
nonsingular variety. Comparing the codimension-1 and 2 terms of this and
other characteristic classes of the discriminant leads to a quick
computation of the degrees of the loci of cuspidal and binodal sections of
a very ample line bundle on a smooth variety, and of the tacnodal locus for
linear systems on a surface. We also compute explicitly all terms in the
Schwartz-MacPherson's classes of strata of the discriminant of cubic plane
curves, and of the discriminants of O(d) on P^1.
-
A remark on the Chern class of a tensor product, with Carel Faber,
Manu. Math. 88 (1995) 85-86.
This ridiculously short note is devoted to the proof of the following
fact: if \alpha is a class of rank r in the Grothendieck group of
vector bundles over a scheme, and L is a line bundle, then
c_{r+1}(\alpha) = c_{r+1}(\alpha\otimes [L]).
The proof is elementary. Maybe the most interesting thing about this
is that it shows up with surprising frequence in intersection-theoretic
computations inspired by enumerative geometry.
-
Singular schemes of hypersurfaces,
Duke Math. Journal, 80 (1995) 325-351.
Let Y be the singular locus of a hypersurface X in a smooth variety M,
with the scheme structure defined by the Jacobian ideal of X (we will
say then that Y is the singular {\it scheme\/} of X, to emphasize that
the scheme structure of Y is important for our considerations). In
this note we consider a class in the Chow group of~Y which arises
naturally in this setup, and which captures much
intersection-theoretic information about the situation. We produce
constraints for a given scheme to be a singular scheme of a
hypersurface, and we obtain applications to duality recovering, and
sometime strengthening, results of Holme, Landman, Ein, and Zak.
The degree of the dimension-0 component of our class agrees with a
known generalization, due to Parusinski, of the usual Milnor number.
-
A blow-up construction and graph coloring,
Discrete Math., 145 (1995) 11-35.
Given a graph G (or more generally a matroid embedded in a projective
space), we construct a sequence of algebraic varieties whose geometry
encodes combinatorial information about G. For example, the chromatic
polynomial of G can be computed as an intersection product of certain
classes on these varieties, or recovered in terms of the Segre classes
of related subschemes of a projective space; other information such as
Crapo's invariant also finds a very natural geometric counterpart.
The note presents this construction, and gives `geometric' proofs of a
number of standard combinatorial results on the chromatic polynomial
and Crapo's invariant.
-
MacPherson's and Fulton's classes of hypersurfaces,
Int. Math. Res. Notices (1994) 455-465.
DOI:10.1155/S1073792894000498
We prove that MacPherson's total Chern class of a singular
hypersurface agrees `numerically' with a class obtained by means of
Fulton's intrinsic class of a scheme. More precisely, for a
hypersurface X with Jacobian scheme J, and any positive integer t,
consider the class P(X,J,t) obtained by taking Fulton's class of the
t-thickening of X along J. Then P(X,J,t) is a polynomial in t (with
coefficients in the Chow group of X), and we show that P(X,J,-1)
agrees with MacPherson's class of X after push-forward via the map
defined by the linear system of X.
We conjecture the equality holds at the level of Chow groups, and
speculate that a similar result should hold for arbitrary algebraic
schemes in characteristic 0.
-
Linear orbits of d-tuples of points in P1, with
Carel Faber,
Jour. für die R. und Ang. Math. 444 (1993), 205-220.
Orbit closures of sets of points of the projective line under the
action of the automorphism group of the latter are studied in terms of
their degree and multiplicity along their boundary.
-
Multiplicities of discriminants, with Fernando Cukierman,
Manu. Math. 78 (1993), 245-258
We compute the multiplicity of the discriminant of a line bundle
L over a nonsingular variety S at a given section X, in terms of
the Chern classes of L and of the cotangent bundle of S, and the
Segre classes of the jacobian scheme of X in S. For S a surface,
we obtain a precise formula that expresses the multiplicity as a
sum of a term due to the non-reduced components of the section, and
a aterm that depends on the Milnor numbers of the singularities of
Xred. Also, under certain hypotheses, we provide
fromulae for the `higher discriminants' that parametrize sections
with a singular point of prescribed multiplicity. As an application,
we obtain criteria for the various discriminants to be `small'.
-
Linear orbits of smooth plane curves, with Carel Faber,
Journal of Alg. Geom. 2 (1993), 155-184.
A desingularization of the orbit closures of plane curves under the
action of the automorphism group of the plane is constructed and used
to study such orbits, obtaining the degree of the orbit closure of
an arbitrary smooth curve in terms of its degree and of the nature
of its flexes and its automorphism group. The result has a transparent
enumerative interpretation.
-
Two characteristic numbers for smooth plane curves of any degree,
Trans. of the Amer. Math. Soc., vol. 329 (1992), 73-96.
We use a sequence of blow-ups over the projective space
parametrizing plane curves of a degree d to obtain some
enumerative results concerning smooth plane curves of arbitrary
degree. For d=4, this gives a first modern verification
of results of H. G. Zeuthen.
-
How many smooth plane cubics with given j-invariant are tangent to
8 lines in general position?,
Enumerative Algebraic Geometry (Copenhagen 1989), Cont. Math.
123 (1991), 15-29.
doi:10.1090/conm/123
We employ a variety of complete cubics to give formulas for
the characteristic numbers of families parametrized by hypersurfaces
F in the P9 of plane cubics, in terms of
infromation easily accessible given the equation of F.
As examples, we obtain explicit results for families of cubics
with given j-invariant and for other families arising naturally
from the geometry of plane cubics.
-
Some characteristic numbers for nodal and cuspidal plane curves of any
degree, Manu. Math. 72 (1991), 425-444.
Two blow-ups over the projective space P^N parametrizing
plane curves of a given degree yield a compactification of the space of
reduced curve used elsewhere to obtain partial enumerative results
for families of nonsingular plane curves. In this paper it is shown
how to employ the construction to obtain enumerative results for
families of plane curves with a node or a cusp. The results recover
known results for cubics, give a first modern verification of some
computation of Zeuthen's for quartics, and are new for higher degree.
The heart of the computation is the derivation of key Segre classes
relating the intersection calculus at the different stages of
the blow-up construction.
-
The enumerative geometry of plane cubics II: nodal and cuspidal cubics,
Math. Annalen 289 (1991), 543-572.
The variety of complete cubics obtained elsewhere is used to
recover classical enumerative results for singular plane cubics,
and obtain new results for cubic curves with prescribed conditions
with respects to flags in the plane.
-
The enumerative geometry of plane cubics I: smooth cubics,
Trans. of the Amer. Math. Soc., vol. 317 (1990), 501-539.
We construct a variety of complete plane cubics by a sequence
of five blow-ups over P9. This enables us to translate
the problem of computing characteristic numbers for a family of
plane cubics into one of computing five Segre classes, and to
recover classic enumerative results of Zeuthen and Maillard.
-
The characteristic numbers of smooth plane cubics,
Algebraic Geometry, Sundance 1986, Springer Lecture Notes 1311,
1-8.
The characteristic numbers for the family of smooth plane cubics
are computed, verifying results of Maillard and Zeuthen.
Quantum Cohomology at the Mittag-Leffler Institute
Fare Matematica. Astratto e concreto nella matematica elementare.
Algebra: Chapter 0.
Proceedings of the AMS Special Session on Singularities and Physics, Knoxville, 2014
Algebra: Notes from the Underground.
Facets of Algebraic Geometry - A Collection in Honor of William Fulton's
80th Birthday, Vol.1
and
Vol.2