Margherita Roggero

Publications (34) View all

• Article: Algebraic Methods for Studying Interactions Between Epidemiological Variables

  F Ricceri, C Fassino, G Matullo, M Roggero, ML Torrente, P Vineis, L Terracini
  Mathematical Modelling of Natural Phenomena 01/2012; · 0.63 Impact Factor
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  Available from: Margherita Roggero

  Article: Borel Degenerations of Arithmetically Cohen-Macaulay curves in P^3

  Gunnar Floystad, Margherita Roggero
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  ABSTRACT: We investigate Borel ideals on the Hilbert scheme components of arithmetically Cohen-Macaulay (ACM) codimension two schemes in P^n. We give a basic necessary criterion for a Borel ideal to be on such a component. Then considering ACM curves in P^3 on a quadric we compute in several examples all the Borel ideals on their Hilbert scheme component. Based on this we conjecture which Borel ideals are on such a component, and for a range of Borel ideals we prove that they are on the component.
  11/2011;
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  Available from: Margherita Roggero

  Article: The locus of points of the Hilbert scheme with bounded regularity

  Edoardo Ballico, Cristina Bertone, Margherita Roggero
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  ABSTRACT: In this paper we consider the Hilbert scheme Hilb_{p(t)}^n parameterizing subschemes of P^n with Hilbert polynomial $p(t)$ and we investigate its locus containing points corresponding to schemes with regularity lower than or equal to a fixed integer r'. This locus is an open subscheme of Hilb_{p(t)}^n and we describe a set of defining equations.
  11/2011;
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  Available from: Margherita Roggero

  Article: Upgraded methods for the effective computation of marked schemes on a strongly stable ideal

  Cristina Bertone, Francesca Cioffi, Paolo Lella, Margherita Roggero
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  ABSTRACT: Let $J\subset S=K[x_0,...,x_n]$ be a monomial strongly stable ideal. The collection $\Mf(J)$ of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S/I$, is called a {\em $J$-marked family}. It can be endowed with a structure of affine scheme, called a {\em $J$-marked scheme}. For special ideals $J$, $J$-marked schemes provide an open cover of the Hilbert scheme $\hilbp$, where $p(t)$ is the Hilbert polynomial of $S/J$. Those ideals more suitable to this aim are the $m$-truncation ideals $\underline{J}_{\geq m}$ generated by the monomials of degree $\geq m$ in a saturated strongly stable monomial ideal $\underline{J}$. Exploiting a characterization of the ideals in $\Mf(\underline{J}_{\geq m})$ in terms of a Buchberger-like criterion, we compute the equations defining the $\underline{J}_{\geq m}$-marked scheme by a new reduction relation, called {\em superminimal reduction}, and obtain an embedding of $\Mf(\underline{J}_{\geq m})$ in an affine space of low dimension. In this setting, explicit computations are achievable in many non-trivial cases. Moreover, for every $m$, we give a closed embedding $\phi_m: \Mf(\underline{J}_{\geq m})\hookrightarrow \Mf(\underline{J}_{\geq m+1})$, characterize those $\phi_m$ that are isomorphisms in terms of the monomial basis of $\underline{J}$, especially we characterize the minimum integer $m_0$ such that $\phi_m$ is an isomorphism for every $m\geq m_0$.
  10/2011;
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  Available from: Margherita Roggero

  Article: Extensors and the Hilbert scheme

  Jerome Brachat, Paolo Lella, Bernard Mourrain, Margherita Roggero
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  ABSTRACT: The Hilbert scheme $\HilbScheme{p(t)}{n}$ parametrizes closed subschemes and families of closed subschemes in the projective space $\PP^n$ with a fixed Hilbert polynomial $p(t)$. It is classically realized as a closed subscheme of a Grassmannian or a product of Grassmannians. In this paper we present a method that allows to derive scheme theoretical global equations for $\HilbScheme{p(t)}{n}$ in the Pl\"ucker coordinates of a Grassmannian $\GrassScheme{p}{N}$, where $p$ and $N$ depend on the dimension $n$ of the projective space and on the Hilbert polynomial $p(t)$. Using this method we obtain the already known set of equations given by Iarrobino and Kleiman in 1999, the one conjectured by Bayer in 1982 and proved by Haiman and Sturmfels in 2004, and also a new set of equations of degree lower than the previous ones. The novelties of our approach are essentially two. The first one is a "local" study of the Hilbert functor through special sets of open subfunctors obtained exploiting the symmetries of the Hilbert scheme and the combinatorial properties of monomial ideals, mainly the Borel-fixed ones. The second one is a generalization of the theory of extensors to the setting of free modules over any ring $A$ and the description of any exterior product of elements of a free submodule in terms of Pl\"ucker coordinates.
  04/2011;