González Pérez, Pedro Daniel

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First Name
Pedro Daniel
Last Name
González Pérez
Universidad Complutense de Madrid
Faculty / Institute
Ciencias Matemáticas
Álgebra, Geometría y Topología
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Now showing 1 - 10 of 28
  • Publication
    Embedded resolutions of non necessarily normal affine toric varieties
    (Académie des Sciences, 2002-03-06) González Pérez, Pedro Daniel; Teissier, Bernard
    We give a method to construct a partial embedded resolution of a nonnecessarily normal affine toric variety Z(Gamma) equivariantly embedded in a normal affine toric variety Z(rho). This partial resolution is an embedded normalization inside a normal toric ambient space and a resolution of singularities of the ambient space, which always exists, provides an embedded resolution. The advantage is that this partial resolution is completely determined by the embedding Z(Gamma) subset of Z(rho). As a by-product, the construction of the normalization is made without an explicit computation of the saturation of the semigroup Gamma of the toric variety (see [3]). This result is valid for a base field k algebraically closed of arbitrary characteristic.
  • Publication
    Resolving singularities of curves with one toric morphism
    (Springer Nature, 2022-11-15) Felipe, Ana Belén de; González Pérez, Pedro Daniel; Mourtada, Hussein
    We give an explicit positive answer, in the case of reduced curve singularities, to a question of B. Teissier about the existence of a toric embedded resolution after reembedding. In the case of a curve singularity pC,Oq contained in a non singular surface S such a reembedding may be defined in terms of a sequence of maximal contact curves associated to C. We prove that there exists a toric modification, after reembedding, which provides an embedded resolution of C. We use properties of the semivaluation space of S at O to describe how the the dual graph of the minimal embedded resolution of C may be seen on the local tropicalization of S associated to this reembedding.
  • Publication
    Quasi-ordinary singularities, essential divisors and Poincaré series
    (Oxford University Press, 2009) González Pérez, Pedro Daniel; Hernando, F.
    We define Poincaré series associated to a germ (S, 0) of toric or analytically irreducible quasiordinary hypersurface singularity, by a finite sequence of monomial valuations such that at least one of them is centered at the point 0. This involves the definition of a multi-graded ring associated to the analytic algebra of the singularity by the sequence of valuations. We prove that the Poincar´e series is a rational function with integer coefficients, which can also be defined as an integral with respect to the Euler characteristic of a function defined by the valuations, over the projectivization of the analytic algebra of the singularity. In particular, the Poincaré series associated to the set of divisorial valuations of the essential divisors, considered both over the singular locus and over the point 0, is an analytic invariant of the singularity. In the quasi-ordinary hypersurface case we prove that this Poincar´e series determines and is determined by the normalized sequence of characteristic monomials. These monomials in the analytic case define a complete invariant of the embedded topological type of the hypersurface singularity.
  • Publication
    Motivic Poincaré series, toric singularities and logarithmic Jacobian ideals
    (American Mathematical Society, 2012) Cobo Pablos, H.; González Pérez, Pedro Daniel
    The geometric motivic Poincare series of a variety, which was introduced by Denef and Loeser, takes into account the classes in the Grothendieck ring of the sequence of jets of arcs in the variety. Denef and Loeser proved that this series has a rational form. We describe it in the case of an affine toric variety of arbitrary dimension. The result, which provides an explicit set of candidate poles, is expressed in terms of the sequence of Newton polyhedra of certain monomial ideals,which we call logarithmic Jacobian ideals, associated to the modules of differential forms with logarithmic poles outside the torus of the toric variety.
  • Publication
    The Semigroup of a Quasi-ordinary Hypersurface
    (Cambridge University Press, 2003-07) González Pérez, Pedro Daniel
    An analytically irreducible hypersurface germ (S, 0) ⊂ (Cd+1, 0) is quasi-ordinary if it canbe defined by the vanishing of the minimal polynomial f ∈ C{X}[Y ] of a fractional power series in the variables X = (X1, . . . , Xd) which has characteristic monomials, generalizing the classical Newton–Puiseux characteristic exponents of the plane-branch case (d = 1). We prove that the set of vertices of Newton polyhedra of resultants of f and h with respect to the indeterminate Y , for those polynomials h which are not divisible by f, is a semigroup of rank d, generalizing the classical semigroup appearing in the plane-branch case.We show that some of the approximate roots of the polynomial f are irreducible quasiordinary polynomials and that, together with the coordinates X1, . . . , Xd, provide a set of generators of the semigroup from which we can recover the characteristic monomials and vice versa. Finally, we prove that the semigroups corresponding to any two parametrizations of (S, 0) are isomorphic and that this semigroup is a complete invariant of the embedded topological type of the germ (S, 0) as characterized by the work of Gau and Lipman.
  • Publication
    Decomposition in bunches of the critical locus of a quasi-ordinary map
    (Cambridge University Press, 2005-03) González Pérez, Pedro Daniel; García Barroso, Evelia Rosa
    A polar hypersurface P of a complex analytic hypersurface germ f = 0 can be investigated by analyzing the invariance of certain Newton polyhedra associated with the image of P, with respect to suitable coordinates, by certain morphisms appropriately associated with f. We develop this general principle of Teissier when f = 0 is a quasi-ordinary hypersurface germ and P is the polar hypersurface associated with any quasi-ordinary projection of f = 0. We show a decomposition of P into bunches of branches which characterizes the embedded topological types of the irreducible components of f = 0. This decomposition is also characterized by some properties of the strict transform of P by the toric embedded resolution of 0 given by the second author. In the plane curve case this result provides a simple algebraic proof of a theorem of Le et al.
  • Publication
    Multiplier ideals of plane curve singularities via Newton polygons
    (2021-09-29) González Pérez, Pedro Daniel; González Villa, Manuel; Guzmán Durán, Carlos R.; Robredo Buces, Miguel
    We give an effective method to determine the multiplier ideals and jumping numbers associated with a curve singularity C in a smooth surface. We characterize the multiplier ideals in terms of certain Newton polygons, generalizing a theorem of Howald, which holds when C is Newton non-degenerate with respect to some local coordinate system. The method uses toroidal embedded resolutions and generating sequences of families of valuations, and can be extended to some classes of higher dimensional hypersurface singularities.
  • Publication
    Ultrametric properties for valuation spaces of normal surface singularities
    (2019) García Barroso, Evelia R.; González Pérez, Pedro Daniel; Popescu-Pampu, Patrick; Ruggiero, Matteo
    Let L be a fixed branch – that is, an irreducible germ of curve – on a normal surface singularity X. If A, B are two other branches, define uL(A, B) := (L · A) (L · B) / A · B , where A · B denotes the intersection number of A and B. Call X arborescent if all the dual graphs of its good resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of Ploski by proving that whenever X is arborescent, the function uL is an ultrametric on the set of branches on X different from L. In the present paper we prove that, conversely, if uL is an ultrametric, then X is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on X, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which uL is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing L to be an arbitrary semivaluation on X and by defining uL on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if X is arborescent, and without any restriction on X we exhibit special subspaces of the space of semivaluations in restriction to which uL is still an ultrametric.
  • Publication
    Una demostración constructiva de un homeomorfismo entre P3 y SO(3)
    (Real Sociedad Matemática Española, 2001) González Pérez, Pedro Daniel; Almira Picazo, J.M.
    The paper gives a very elementary proof of the well-known result that the projective space P3 is homeomorphic to the special orthogonal Lie group SO(3). The proof uses only linear algebra, elementary topology and elementary affine geometry. One of its motivations is the theory developed in the classical book of M. L. Curtis [Matrix groups, Springer, New York, 1979; MR0550439 (81c:22001)]. The paper is also very useful for young students in mathematics to become initiated in Lie theory and its applications.
  • Publication
    Multi-Harnack smoothings of real plane branches
    (Société Mathématique de France, 2010) González Pérez, Pedro Daniel; Risler, Jean-Jacques
    This paper is motivated by the results of G. Mikhalkin about a certain class of real algebraic curves, called Harnack curves, in toric surfaces. Mikhalkin has proved the existence of such curves as well as topological uniqueness of their real locus. The authors are concerned about an analogous statement in the case of a smoothing of a real plane branch (C, 0) _ (C2, 0) (an analytically irreducible germ of a real curve). They introduce the class of multi-Harnack smoothings of (C, 0) and prove its existence along with its topological uniqueness. Theorem 9.3. Any real plane branch (C, 0) has a multi-Harnack smoothing. Theorem 9.4. Let (C, 0) be a real branch. The topological type of multi-Harnack smoothings of (C, 0) is unique. There are at most two signed topological types of multi-Harnack smoothings of (C, 0). These types depend only on the sequence {(nj ,mj)}, which determines and is determined by the embedded topological type of (C, 0) _ (C2, 0). In terms of the parameters, multi-Harnack smoothings are multi-semi-quasi-homogeneous, which lets the authors analyze also the asymptotic multi-scales of the ovals.