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Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs

Victor V. Batyrev (1999)

Journal of the European Mathematical Society

Using non-Archimedian integration over spaces of arcs of algebraic varieties, we define stringy Euler numbers associated with arbitrary Kawamata log-terminal pairs. There is a natural Kawamata log-terminal pair corresponding to an algebraic variety V having a regular action of a finite group G . In this situation we show that the stringy Euler number of this pair coincides with the physicists’ orbifold Euler number defined by the Dixon-Harvey-Vafa-Witten formula. As an application, we prove a conjecture...

Note on the Newton number.

Gwoździewicz, Janusz (2008)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

On actions of ℂ * on algebraic spaces

Andrzej Bialynicki-Birula (1993)

Annales de l'institut Fourier

The main result of the paper says that all schematic points of the source of an action of C * on an algebraic space X are schematic on X .

On equivalences of derived and singular categories

Vladimir Baranovsky, Jeremy Pecharich (2010)

Open Mathematics

Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → 𝔸 1 , g:Y → 𝔸 1 . Assuming that there exists a complex of sheaves on X × 𝔸 1 Y which induces an equivalence of D b(X) and D b(Y), we show that there is also an equivalence of the singular derived categories of the fibers f −1(0) and g −1(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective...

On geometry of binary symmetric models of phylogenetic trees

Weronika Buczyńska, Jaroslaw A. Wiśniewski (2007)

Journal of the European Mathematical Society

We investigate projective varieties which are binary symmetric models of trivalent phylogenetic trees. We prove that they have Gorenstein terminal singularities and are Fano varieties of index 4 and dimension equal to the number of edges of the tree in question. Moreover any two such varieties which are of the same dimension are deformation equivalent, that is, they are in the same connected component of the Hilbert scheme of the projective space. As an application we provide a simple formula for...

On higher dimensional Hirzebruch-Jung singularities.

Patrick Popescu-Pampu (2005)

Revista Matemática Complutense

A germ of normal complex analytical surface is called a Hirzebruch-Jung singularity if it is analytically isomorphic to the germ at the 0-dimensional orbit of an affine toric surface. Two such germs are known to be isomorphic if and only if the toric surfaces corresponding to them are equivariantly isomorphic. We extend this result to higher-dimensional Hirzebruch-Jung singularities, which we define to be the germs analytically isomorphic to the germ at the 0-dimensional orbit of an affine toric...

On orbits of the automorphism group on an affine toric variety

Ivan Arzhantsev, Ivan Bazhov (2013)

Open Mathematics

Let X be an affine toric variety. The total coordinates on X provide a canonical presentation X ¯ → X of X as a quotient of a vector space X ¯ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.

On some numerical properties of Fano varieties

Cinzia Casagrande (2004)

Bollettino dell'Unione Matematica Italiana

This is the text of a talk given at the XVII Convegno dell’Unione Matematica Italiana held at Milano, September 8-13, 2003. I would like to thank Angelo Lopez and Ciro Ciliberto for the kind invitation to the conference. I survey some numerical conjectures and theorems concerning relations between the index, the pseudo-index and the Picard number of a Fano variety. The results I refer to are contained in the paper [3], wrote in collaboration with Bonavero, Debarre and Druel.

On the graph labellings arising from phylogenetics

Weronika Buczyńska, Jarosław Buczyński, Kaie Kubjas, Mateusz Michałek (2013)

Open Mathematics

We study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.

On the Łojasiewicz exponent of the gradient of a polynomial function

Andrzej Lenarcik (1999)

Annales Polonici Mathematici

Let h = ∑ h α β X α Y β be a polynomial with complex coefficients. The Łojasiewicz exponent of the gradient of h at infinity is the least upper bound of the set of all real λ such that | g r a d h ( x , y ) | ≥ c | ( x , y ) | λ in a neighbourhood of infinity in ℂ², for c > 0. We estimate this quantity in terms of the Newton diagram of h. Equality is obtained in the nondegenerate case.

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