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Paolo Aluffi

Publications and source records attributed to Paolo Aluffi.

At least 55 records · Page 3Linked to original sources
arXiv2009

Limits of PGL(3)-translates of plane curves, II

Every complex plane curve C determines a subscheme S of the $P^8$ of 3x3 matrices, whose projective normal cone (PNC) captures subtle invariants of C. In "Limits of PGL(3)-translates of plane curves, I" 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" of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, an invariant of natural enumerative significance.

math.AG↗
arXiv2009

Parametric Feynman integrals and determinant hypersurfaces

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.

math.AG↗
arXiv2008

Algebro-geometric Feynman rules

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.

hep-th↗
arXiv2008

Chern classes of blow-ups

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 the 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).

math.AG↗
arXiv2008

Feynman motives of banana graphs

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.

hep-th↗
arXiv2007

Chern class identities from tadpole matching in type IIB and F-theory

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.

hep-th↗
arXiv2006

Chern classes of Schubert cells and varieties

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 $d$.

math.AG↗
arXiv2006

Limits of Chow groups, and a new construction of Chern-Schwartz-MacPherson classes

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.

math.AG↗
arXiv2006

Classes de Chern des varietes singulieres, revisitees

We introduce a notion of `proChow group' of varieties, agreeing with the notion of Chow group for complete varieties and covariantly functorial with respect to 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: Kwiecinski's product formula, and the Ehlers-Barthel-Brasselet-Fieseler computation of Chern-Schwartz-MacPherson classes of toric varieties.

math.AG↗
arXiv2005

Celestial integration, stringy invariants, and Chern-Schwartz-MacPherson classes

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.

math.AG↗
arXiv2004

Chern classes of birational varieties

A theorem of Batyrev's asserts that if two nonsingular varieties V,W are birational, and their canonical bundles agree after pull-back to a resolution of indeterminacies of a birational map between them, then the Betti numbers of V and W coincide. We prove that, in the same hypotheses, the total homology Chern classes of V and W are push-forwards of the same class in the Chow group of the resolution. 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.

math.AG↗
arXiv2004

Modification systems and integration in their Chow groups

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. v2: References added, and overly optimistic claims concerning non log-terminal singularities expunged.

math.AG↗
arXiv2003

Limits of translates of plane curves -- on a paper of Aldo Ghizzetti

We study the limits of PGL(3) translates of an arbitrary plane curve, giving a description of all possible limits of a given curve and computing the multiplicities of corresponding components in the normal cone to the base scheme of a related linear system. This information is a key step in the computation of the degree of the closure of the linear orbit of an arbitrary plane curve. Our analysis recovers and extends results obtained by Aldo Ghizzetti in the 1930's.

math.AG↗
arXiv2003

Shadows of blow-up algebras

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, corresponding to the identity, and the `quasi-symmetric blow-up', corresponding to an embedding 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 is 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.

math.AG↗
arXiv2002

Computing characteristic classes of projective schemes

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.

math.AG↗
arXiv2002

Inclusion-exclusion and Segre classes, II

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 math.AG/0203122, in certain special (but representative) cases.

math.AG↗
arXiv2002

Inclusion-exclusion and Segre classes

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 a scheme in terms of this class.

math.AG↗