-
Grothendieck classes and Chern classes of hyperplane arrangements,
arXiv:1103.2777, to appear in IMRN.
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, arXiv:1009.2483.
To appear in Arkiv foer Matematik.
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.
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) 1-53.
DOI: 10.1.1.145.2590
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.
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 pour variétés singulières,
revisitées,
(French and English versions)
C. R. Math. Acad. Sci. Paris 342 (2006), no. 6, 405--410.
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.
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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.
Let $\varphi: V\dashrightarrow W$ 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 $\varphi$.
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.
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.
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.