186 research outputs found
Plane curves with small linear orbits I
The `linear orbit' of a plane curve of degree d is its orbit in the
projective space of dimension d(d+3)/2 parametrizing such curves 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].Comment: 34 pages, 4 figures, AmS-TeX 2.1, requires xy-pic and eps
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.Comment: 8 page
Verdier specialization via weak factorization
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.Comment: Minor revision. To appear in Arkiv f\"or Matemati
On generalized Sethi-Vafa-Witten formulas
We present a formula for computing proper pushforwards of classes in the Chow
ring of a projective bundle under the projection \pi:\Pbb(\Escr)\rightarrow
B, for a non-singular compact complex algebraic variety of any dimension.
Our formula readily produces generalizations of formulas derived by Sethi,Vafa,
and Witten to compute the Euler characteristic of elliptically fibered
Calabi-Yau fourfolds used for F-theory compactifications of string vacua. The
utility of such a formula is illustrated through applications, such as the
ability to compute the Chern numbers of any non-singular complete intersection
in such a projective bundle in terms of the Chern class of a line bundle on
Graph hypersurfaces and a dichotomy in the Grothendieck ring
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.Comment: 8 pages, LaTe
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.Comment: 26 pages, LaTeX, 1 figur
Likelihood Geometry
We study the critical points of monomial functions over an algebraic subset
of the probability simplex. The number of critical points on the Zariski
closure is a topological invariant of that embedded projective variety, known
as its maximum likelihood degree. We present an introduction to this theory and
its statistical motivations. Many favorite objects from combinatorial algebraic
geometry are featured: toric varieties, A-discriminants, hyperplane
arrangements, Grassmannians, and determinantal varieties. Several new results
are included, especially on the likelihood correspondence and its bidegree.
These notes were written for the second author's lectures at the CIME-CIRM
summer course on Combinatorial Algebraic Geometry at Levico Terme in June 2013.Comment: 45 pages; minor changes and addition
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