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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X=X_1\cup\cdots\cup X_s\) be a reducible projective curve such that each \(X_i\) is smooth and \(f,u:X\to\mathbb P^1\) finite morphisms. Here we give numerical conditions which implies that ``\(u\) is partially composed with \(f\)''. reducible curves; Castelnuovo-Severi inequality Singularities of curves, local rings, Special divisors on curves (gonality, Brill-Noether theory) Composed maps for reducible curves with smooth components | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A geometrical proof of the Frenkel-Kac-Segal isomorphism is given. In the following, \(G\) will be always a simple, connected algebraic group over \(\mathbb{C}\) with Lie algebra \(\mathfrak{g}\). Let \(\hat{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus \mathbb{C}K\) be the associated untwisted affine Kac-Moody algebra. Given a positive integer \(k\), let \(\mathbb{V}(k\Lambda)=Ind_{\mathfrak{g}\otimes\mathbb{C}[t]\oplus \mathbb{C}K}^{\hat{\mathfrak{g}}}\mathbb{C}\) be the level \(k\) module of \(\hat{\mathfrak{g}}\) and let \(\mathbb{L}(k\Lambda)\) be its unique irreducible quotient; \(\mathbb{V}(k\Lambda)\) and \(\mathbb{L}(k\Lambda)\) have a structure of vertex algebra. Let \(\mathfrak{t}\subset\mathfrak{g}\) be a Cartan algebra and let \(\hat{\mathfrak{t}}\subset\hat{\mathfrak{g}}\) be the associated Heisenberg Lie algebras. Let \(V_{R_{G}}\) be \(\bigoplus_{\lambda\in R_{G}}\pi_{\lambda}\) where \(R_{G}\) is the coroot lattice of \(\mathfrak{g}\) and \(\pi_{\lambda}\) is the Fock module of \(\mathfrak{t}\) with highest weight \(\iota\lambda\). If \(\mathfrak{g}\) is simply-laced, then \(V_{R_{G}}\) have a structure of vertex algebras and the Frenkel-Kac-Segal states that, given a simple-laced simple algebra \(\mathfrak{g}\), (1) \( V_{R_{G}}\cong\mathbb{L}(\Lambda)\) as vertex algebras. In particular, they are isomorphic as \(\hat{\mathfrak{t}}\)-modules.
The author consider the following geometrical interpretation. Let \(Gr_{G}=\mathcal{G}_{\mathcal{K}}/\mathcal{G}_{\mathcal{O}}\) be the affine Grassmannian of \(G\), where \(\mathcal{G}_{\mathcal{K}}\) is the group of maps from the punctured disc to \(G\) and \(\mathcal{G}_{\mathcal{O}}\) is the group of maps from the disc to \(G\). If \(G\) is simply-connected then the Picard group of \(Gr_{G}\) is generated by an ample invertible sheaf \(\mathcal{L}_{G}\). Then the Borel-Weyl theorem for affine Kac-Moody algebra identifies \(\mathbb{L}(k\Lambda)\) (resp. \(V_{R_{G}}\)) with \(H^{0}(Gr_{G},\mathcal{L}_{G}^{\otimes k})^{*}\) (resp. \(H^{0}(Gr_{G},\mathcal{O}_{Gr_{T}}\otimes\mathcal{L}_{G})^{*}\)). Thus \(\mathbb{L}(\Lambda)\cong V_{R_{G}}\) as \(\hat{\mathfrak{t}}\)-module if and only if the restriction morphism \(\varphi:\mathcal{L}_{G}\rightarrow \mathcal{O}_{Gr_{T}}\otimes \mathcal{L}_{G}\) induces an isomorphism between the spaces of global sections. If \(G\) is not simple connected, then \(Gr_{G}\) is not connected and the author define \(\mathcal{L}_{G}\) as the line bundle whose restriction to each connected component is the ample generator of its Picard group. The Borel-Weyl theorem hold again (see Proposition 1.4.4). Recall that \(Gr_{G}\) is stratified by \(G_{\mathcal{O}}\)-orbits, \(\{Gr^{\lambda}_{G}\}\), indexed by the dominant coweights \(\{\lambda\}\). Moreover, \(Gr_{G}=\displaystyle{\lim_\rightarrow}\, \overline{Gr}^{\lambda}_{G}\), where the Schubert varieties \(\overline{Gr}^{\lambda}_{G}\) are defined as the closures of the \(Gr^{\lambda}_{G}\).
The maximal torus \(T\) of \(G\) acts on the Schubert varieties and the natural embedding \(Gr_{T}\subset Gr_{G}\) identifies \(Gr_{T}\) with the \(T\)-fixed point scheme of \(Gr_{G}\) (see \S1.3). Moreover \(Gr_{T}\times_{Gr_{G}}\overline{Gr}^{\lambda}_{G}\) is the \(T\)-fixed point scheme of \(\overline{Gr}^{\lambda}_{G}\). The main theorem of this article states that the restriction of \(\varphi\) to \(\overline{Gr}^{\lambda}_{G}\) induces an isomorphism on the global sections, (2) \( H^{0}(\overline{Gr}^{\lambda}_{G},\mathcal{L}_{G})\rightarrow H^{0}(\overline{Gr}^{\lambda}_{G},\mathcal{O}_{(\overline{Gr}^{\lambda}_{G})^{T}}\otimes\mathcal{L}_{G})\), if \(G\) has type A or D. Furthermore, the same fact holds for many coweights if \(G\) has type E. In many parts of the proof it is used that \((\overline{Gr}^{\lambda}_{G})^{T}\) is a finite scheme.
The difficult part of the proof is showing the injectivity of (2). Indeed it is proved that the restriction of \(\mathcal{L}_{G}^{k}\) from \(\overline{Gr}^{\lambda}_{G}\) to \((\overline{Gr}^{\lambda}_{G})^{T}\) induces a surjective morphism on the global sections for any simple algebraic group \(G\).
The first step to prove the injectivity is a reduction to the case of fundamental dominant coweights. Given two dominant coweights \(\lambda\) and \(\mu\), the author constructs a family of varieties with generic fibre is \(\overline{Gr}^{\lambda}_{G}\times \overline{Gr}^{\mu}_{G}\) while the special fibre is \(\overline{Gr}^{\lambda+\mu}_{G}\), using the following result on the Demazure affine modules (here \(\widetilde{G}\) is the simply connected cover of \(G\)): \(H^{0}(\overline{Gr}^{\lambda+\mu}_{G},\mathcal{L}_{G}^{k})\cong H^{0}(\overline{Gr}^{\lambda}_{G},\mathcal{L}_{G}^{k}) \otimes H^{0}(\overline{Gr}^{ \mu}_{G},\mathcal{L}_{G}^{k})\) as \(\widetilde{G}\)-modules. The author includes a proof of this isomorphism which is more geometrical than the original one. To prove the reduction step, the author uses the this family together with the interpretation of the affine Grassmannian as a module space over a smooth curve.
Next, he prove that the Schubert variety \(\overline{Gr}_{G}^{\lambda}\) contains many rational curve with known degree. If \(\lambda\) is minuscule (and \(G\) is simply laced) then these curves have degree one. The author proves the injectivity by showing that an arbitrarily fixed section of \(H^{0}(\overline{Gr}_{G}^{\lambda},\mathcal{I}^{\lambda}(1))\) is zero over certain \(T\)-invariant subvarieties \(Z\) (by induction on the dimension of \(Z\)). Here \(\mathcal{I}^{\lambda}\subset\mathcal{O}_{\overline{Gr}_{G}^{\lambda}}\) is the ideal sheaf defining \((\overline{Gr}_{G}^{\lambda})^{T}\). Moreover, the previous subvarieties includes all the \(T\)-invariant curves and the whole Schubert variety. Therefore, the case of type A is proved.
Next, he prove the injectivity when \(\lambda\) is a longest root. The proof is similar, but he need to consider also curve of degree 2. Moreover he uses the result for \(G=SL_{2}\). This fact proves the case of \(D_{4}\).
The case of \(D_{n}\) is proved by induction on \(n\) and by induction on \(i\), where \(\lambda\) is \(i\)-th fundamental coweight and the weight are indexed according to [\textit{N. Bourbaki}, Groupes et algébres de Lie. Chapitres 4, 5 et 6. Elements de Mathematique. (Paris) etc.: Masson. (1981; Zbl 0483.22001)]. He need also the isomorphism (2) for the case A. Finally, the author can prove some other cases when the type of \(G\) is \(E\) and he conjectures that the map is always injective for the type \(E\).
It is necessary to note that the author can reprove the FKS-isomorphism for all the simply laced group. The isomorphism (1) as \(\hat{\mathfrak{t}}\)-modules clearly follows from (2) for the type A and D. To prove the case \(E\) the author show that also in this case any connected components of \(Gr_{G}\) is the direct limits of Schubert varieties associated to weight for which (2) is an isomorphism.
To prove that (1) is an isomorphism of vertex algebras he uses the languages of Kac-moody factorization algebras. Finally, he prove an identification of the modules over \(V_{R_{G}}\) with the modules over \( \mathbb{L}(\Lambda)\). basic representation; Frenkel-Kac-Segal isomorphism; affine Grassmannian Zhu, X., \textit{affine Demazure modules and \textit{T}-fixed point subschemes in the affine Grassmannian}, Adv. Math., 221, 570-600, (2009) Grassmannians, Schubert varieties, flag manifolds, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Vertex operators; vertex operator algebras and related structures Affine Demazure modules and \(T\)-fixed point subschemes in the affine Grassmannian | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We improve the algorithms of \textit{A. G. B. Lauder} and \textit{D. Wan} [in: Buhler, J. P. (ed.) et al., Algorithmic number theory. Lattices, number fields, curves and cryptography. Cambridge: Cambridge University Press. Mathematical Sciences Research Institute Publications 44, 579--612 (2008; Zbl 1188.11069)] and \textit{D. Harvey} [Proc. Lond. Math. Soc. (3) 111, No. 6, 1379--1401 (2015; Zbl 1333.11062)] to compute the zeta function of a system of \(m\) polynomial equations in \(n\) variables, over the \(q\) element finite field \(\mathbb{F}_q\), for large \(m\). The dependence on \(m\) in the original algorithms was exponential in \(m\). Our main result is a reduction of the dependence on \(m\) from exponential to polynomial. As an application, we speed up a doubly exponential algorithm from a recent software verification paper [\textit{G. Barthe} et al., Proceedings of the 2020 35th annual ACM/IEEE symposium on logic in computer science, LICS 2020. New York, NY: Association for Computing Machinery (ACM), 155--166 (2020; Zbl 1496.68186)] (on universal equivalence of programs over finite fields) to singly exponential time. One key new ingredient is an effective, finite field version of the classical Kronecker theorem which (set-theoretically) reduces the number of defining equations for a polynomial system over \(\mathbb{F}_q\) when \(q\) is suitably large. counting solutions; finite fields; program verification Number-theoretic algorithms; complexity, Varieties over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Specification and verification (program logics, model checking, etc.) Computing zeta functions of large polynomial systems over finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We use basic algebraic topology and Ellingsrud-Strømme results on the Betti numbers of punctual Hilbert schemes of surfaces to compute a generating function for the Euler characteristic numbers of the Douady spaces of ''\(n\)-points'' associated with a complex surface. The projective case was first proved by \textit{L. Göttsche} [Math. Ann. 286, No. 1-3, 193--207 (1990; Zbl 0679.14007)]. Mark Andrea A. de Cataldo, Hilbert schemes of a surface and Euler characteristics, Arch. Math. (Basel) 75 (2000), no. 1, 59 -- 64. Parametrization (Chow and Hilbert schemes), Complex-analytic moduli problems Hilbert schemes of a surface and Euler characteristics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the preview in Zbl 0714.13017. 2-dimensional local domain; finite local cohomology module; blowing-up; Cohen-Macaulayness of singularity Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Local cohomology and commutative rings, Global theory and resolution of singularities (algebro-geometric aspects), Multiplicity theory and related topics, Integral domains Toward parametric Cohen-Macaulayfication of two-dimensional finite local cohomology domains. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider the topology induced by Hausdorff distance on the projective subvarieties of \(\mathbb P^m(\mathbb C)\). We construct the minimal stratification, for this topology, of the space of coefficients of a homogeneous polynomial system with parameters. We give an algorithmic description of this stratification based on some usual algorithms in computer algebra such as equidimensional decomposition or normalization of a projective variety and also on a not so usual one, the fiber power of a morphism. The input algorithmic problem is an algebraic question in \(\mathbb Q\) all the coefficients of the intermediate polynomials that we will consider are algebraic numbers. Our methods of proof of theorems, however, rely on analytic geometric properties. Computational aspects of higher-dimensional varieties, Local structure of morphisms in algebraic geometry: étale, flat, etc., Topological properties in algebraic geometry Continuity loci for polynomial systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show how to use numerical continuation to compute the intersection \(C=A\cap B\) of two algebraic sets \(A\) and \(B\), where \(A, B\), and \(C\) are numerically represented by witness sets. En route to this result, we first show how to find the irreducible decomposition of a system of polynomials restricted to an algebraic set. The intersection of components \(A\) and \(B\) then follows by considering the decomposition of the diagonal system of equations \(u-v = 0\) restricted to \(\{u,v\} \in A \times B\). An offshoot of this new approach is that one can solve a large system of equations by finding the solution components of its subsystems and then intersecting these. It also allows one to find the intersection of two components of the two polynomial systems, which is not possible with any previous numerical continuation approach. components of solutions; embedding; generic points; homotopy continuation; irreducible components; numerical algebraic geometry; polynomial system Sommese, A.J.; Verschelde, J.; Wampler, C.W., Homotopies for intersecting solution components of polynomial systems, SIAM J. Numer. Anal., 42, 1552-1571, (2004) Polynomials, factorization in commutative rings, Computational aspects in algebraic geometry, Numerical computation of solutions to systems of equations, Symbolic computation and algebraic computation Homotopies for intersecting solution components of polynomial systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Hilbert scheme \(X^{[n]}\) of points in a smooth projective surface \(X\) is a desingularization of the \(n\)-th symmetric product of \(X\). An element \(\xi \in X^{[n]}\) is a length-\(n\) \(0\)-dimensional closed subscheme of \(X\).
In the paper under review the authors study curves contained in the Hilbert scheme \(X^{[n]}\) when \(X\) is a simply-connected smooth projective surface for the case \(n \geq 2\). For a fixed point \(x \in X\), let \(M_2(x)=\{\xi\in X^{[2]} \mid {\text{supp}}(\xi)=\{x\}\}\). It is known that \(M_2(x)\cong \mathbb{P}^1\). Let
\[
\beta_n =\{\xi+x_1+\ldots+x_{n-2}\in X^{[n]} \mid \xi\in M_2(x)\} \tag{1}
\]
for fixed distinct points \(x, x_1,\ldots,x_{n-2}\). Using well-known results about Hilbert schemes, the authors characterize all the curves \(\gamma\) in the Hilbert scheme \(X^{[n]}\) homologous to \(\beta_n\). It turns out that the moduli space \(\mathcal{M}(\beta_n)\) of all these curves has dimension \((2n-2)\) and its top stratum consists of all the curves \(\gamma\) of the form (1). Moreover they determine the normal bundles of the curves \(\gamma\) of the form (1). Now let \(X=\mathbb{P}^2\). For a fixed line \(\ell \subset X\) and distinct points \(x_1, \ldots, x_{n-1} \in X\) such that \(x_i \notin \ell\), where \(i=1,\ldots, (n-1)\), let \(\beta_{\ell}\) be the curve in \(X^{[n]}\) of the form
\[
\beta_{\ell} =\{x+x_1+\ldots+x_{n-1}\in X^{[n-1]} \mid x \in \ell\} \tag{2}
\]
The authors prove that the effective cone of the Hilbert scheme \(X^{[n]}\) is spanned by \(\beta_n\) and \(\beta_{\ell}-(n-1)\beta_n\), and determine the nef cone of \(X^{[n]}\). In addition, they show that a curve \(\gamma\) in \(X^{[n]}\) is homologous to \(\beta_{\ell}-(n-1)\beta_n\) if and only if there exists a line \(C\) in \(X=\mathbb{P}^2\) such that \(\gamma\) is a line in \(C^{[n]}\subset X^{[n]}\) (Theorem 5.1). So all these curves \(\gamma\) are parametrized by the Grassmannian bundle over the dual space \((\mathbb{P}^2)^{*}\).
Finally, the authors remark that results of the paper may be used to compute the \(1\)-point Gromov-Witten invariants in the Hilbert scheme \(X^{[n]}\), and to study the quantum cohomology of \(X^{[n]}\). simply-connected smooth projective surface; algebraic curve; normal bundle; homology group Li W.-P., Qin Z.B., Zhang Q.: Curves in the Hilbert schemes of points on surfaces. Contemp. Math. 322, 89--96 (2003) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Curves in the Hilbert schemes of points on surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0653.00008.]
Let Q be a smooth complex quadric of dimension n-1 and let \(Z^*(d,g,n,Q)\) be the subset of the Hilbert scheme of Q consisting of the smooth nondegenerated connected curved of degree d and genus g. The author proves that if \(n\geq 7\) and \(g\leq (n/2)-1\) then \(Z^*(d,g,n,Q)\) is smooth and irreducible. If \(d>n\), a general element of the closure of \(Z^*(d,0,n,Q)\) in the Hilbert scheme is shown to have maximal rank. deformation; degeneration; complex quadric; Hilbert scheme Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry On the Hilbert scheme of curves in a smooth quadric | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal C\) be a category with pullbacks and \(\mathcal A\) a \(\mathcal C\)-indexed category, i.e. a pseudo-functor from \(\mathcal C\) into the \(2\)-category of categories. To each morphism \(f\colon c \to c'\) one can associate a category \(\mathcal Des_{\mathcal A}(f)\) of descent data relative to \(f\). Furthermore there are canonical descent data for each \(a \in \mathcal A(c)\) given by a functor \(K^f_{\mathcal A}\colon \mathcal A(c) \to {\mathcal Des_{\mathcal A}(f)}\) [see this text or, e.g. \textit{G.~Janelidze} and \textit{W.~Tholen}, Appl.~Categ.~Struct.~2, No. 3, 245--281 (1994; Zbl 0805.18005)]. In case the functor \(K^f_{\mathcal A}\) is faithful (\(k = 0\)), fully faithful (\(k = 1\)) or an equivalence of categories (\(k = 2\)), one calls \(f\) an \(\mathcal A\)-\(k\)-descent morphism [\textit{A.~Grothendieck}, Sem.~Bourbaki 12, No. 190 (1960; Zbl 0229.14007)]. The morphism \(f\) is called a stable \(\mathcal A\)-\(k\)-descent morphism if its pullback along any morphism in \(\mathcal C\) is an \(\mathcal A\)-\(k\)-descent morphism.
Let \(\mathcal C\) be the category of schemes and \(\mathcal A\) the \(\mathcal C\)-indexed category of quasi-coherent modules, i.e. for every scheme \(c \in \mathcal C\) let \(\mathcal A(c)\) be the category quasi-coherent modules on \(c\). Then a classical result by \textit{A. Grothendieck} [loc. cit.] says that every faithfully flat, quasi-compact morphism \(f\) of schemes is a stable \(\mathcal A\)-\(2\)-descent morphism. The converse is not true as pointed out in this article.
The purpose of this paper is to generalise Grothendieck's result to have a complete description of the class of quasi-compact morphisms of schemes which are stable \(\mathcal A\)-\(2\)-descent morphisms. In order to do so, the author introduces the notion of a pure morphism of schemes giving back the usual notion of a pure morphism between commutative rings in the affine case, which is one of the main achievements of the article. The definition is as follows: Let \(\mathcal M\) be the class of closed immersions in the category of schemes and let \(\mathcal E\) be the class of morphisms \(f\) of schemes such that for each commutative square \(gp = qf\) with \(g \in \mathcal M\) there is a unique diagonal morphism \(h\) with \(h f = p\) and \(g h = q\). (It follows that \((\mathcal E, \mathcal M)\) is a factorisation system on the category of schemes [\textit{J. Adámek, H.~Herrlich} and \textit{G.~E.~Strecker}, Abstract and concrete categories. The joy of cats. (Wiley-Interscience Publication. New York) (1990; Zbl 0695.18001)].) Then a morphism \(f\) of schemes is pure if it is the pullback of a morphism of \(\mathcal E\).
With this definition one can finally state one of the main results of the text:
A quasi-compact morphism \(f\) of schemes is a stable \(\mathcal A\)-\(k\)-morphism (\(k \in \{0, 1, 2\}\)) if and only if \(f\) is pure.
The author also derives analoguous results for the \(\mathcal C\)-indexed category of quasi-coherent algebras.
It should be emphasised that the author clearly separates the notions and statements one needs from commutative algebra and algebraic geometry from purely categorical notions and statements when he introduces new notions or states his lemmas and theorems. This should make any generalisation of the results one may be tempted to do much easier. scheme; quasi-coherent module; indexed category; (effective) descent morphism; pure morphism of schemes B. Mesablishvili, ''Descent theory for schemes,'' Appl. Categ. Struct., 12, Nos. 5--6, 485--512 (2004). Schemes and morphisms, Local structure of morphisms in algebraic geometry: étale, flat, etc., Epimorphisms, monomorphisms, special classes of morphisms, null morphisms, Special properties of functors (faithful, full, etc.), Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.), Eilenberg-Moore and Kleisli constructions for monads Descent theory for schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study affine group schemes over a discrete valuation ring \(R\) using two techniques: Neron blowups and Tannakian categories. We employ the theory developed to define and study differential Galois groups of \(\mathcal{D}\)-modules on a scheme over a \(R\). This throws light on how differential Galois groups of families degenerate. Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Group schemes, Linear algebraic groups over adèles and other rings and schemes On the structure of affine flat group schemes over discrete valuation rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study non-degenerate subschemes \(Y\subseteq\mathbb{P}^n\) with degenerate general hyperplane section. Our results consist in some steps towards a classification of such schemes \(Y\) in arbitrary characteristic when \(Y\) is locally a Cohen-Macaulay curve (with more details for \(n=4,5)\) and in characteristic zero when \(Y\) is an \(S_2\) equidimensional scheme of dimension \(\geq 2\). Our main results are proved for connected schemes. \(S_2\) equidimensional scheme; degenerate general hyperplane section; Cohen-Macaulay curve Ballico, E.; Chiarli, N.; Greco, S.: Projective schemes with degenerate general hyperplane section. Beiträge algebra geom. 40, 565-576 (1999) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Plane and space curves, Projective techniques in algebraic geometry Projective schemes with degenerate general hyperplane section | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we will investigate further properties of \(\mathcal A\)-schemes introduces in [Tak]. The category of \(\mathcal A\)-schemes possesses many properties of the category of coherent schemes, and in addition, it is co-complete and complete. There is the universal compactification, namely, the Zariski-Riemann space in the category of \(\mathcal A\)-schemes. We compare it with the classical Zariski-Riemann space, and characterize the latter by a left adjoint. Generalizations (algebraic spaces, stacks) \(\mathcal A\)-schemes and Zariski-Riemann spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show that the sheaf of \(\mathbb{A}^1\)-connected components of a Nisnevich sheaf of sets and its universal \(\mathbb{A}^1\)-invariant quotient (obtained by iterating the \(\mathbb{A}^1\)-chain connected components construction and taking the direct limit) agree on field-valued points. This establishes an explicit formula for the field-valued points of the sheaf of \(\mathbb{A}^1\)-connected components of any space. Given any natural number \(n\), we construct an \(\mathbb{A}^1\)-connected space on which the iterations of the naive \(\mathbb{A}^1\)-connected components construction do not stabilize before the \(n\)-th stage. \(\mathbb{A}^1\)-connected components; \(\mathbb{A}^1\)-chain connected components; Morel's conjecture Motivic cohomology; motivic homotopy theory Remarks on iterations of the \(\mathbb{A}^1\)-chain connected components construction | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We deal with the classification of semisimple group schemes via the Bruhat-Tits' presentation of non-abelian cohomology. The goal is to generalize Galois techniques to more general rings. It leads us to investigate the concept of reducibility for reductive group schemes with special attention to one parameter subgroups. It requires also the study of parabolic subgroups and their normalizers of a not-necessarily connected affine smooth group scheme G whose neutral component \(G^0\) is reductive. The text discusses in an appendix also certain analogies for Weyl group schemes and twisted root data. Gille, P., Sur la classification des schémas en groupes semi-simples, ``Autour des schémas en groupes, III'', Panor. Synthèses, 47, 39-110, (2015) Group schemes, Group actions on varieties or schemes (quotients) On the classification of semisimple group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems From the abstract: Let \(X\) denote a fixed smooth projective curve of genus \(1\), defined over an algebraically closed field of characteristic different from \(2\). For any positive integer \(n\), we will consider the moduli space \(H(X,n)\) of degree \(n\) finite separable covers of \(X\) by a hyperelliptic curve marked at a triplet of Weierstrass points. We start parametrizing \(H(X,n)\) by a suitable space of rational fractions, obtaining a polynomial characterization of those having order of osculation \(d \geq 1\). We deduce systems of polynomial equations, whose set of solutions codifies all degree \(n\) hyperelliptic \(d\) osculating covers of \(X\). Weierstrass points; hyperelliptic curves; Abel map Coverings of curves, fundamental group Systems of polynomial equations for hyperelliptic \(d\)-osculating covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a principally polarized complex abelian variety \((T,L)\), the author studies the closures \({\mathcal M}_k\) of the moduli space of \(\mu\)-stable vector bundles in the moduli space of \(S\)-equivalence classes of \(G\)- semistable torsion-free sheaves with Chern character \((2,0, - k)\). If \(T\) is irreducible, he proves that \({\mathcal M}_2\) is naturally fibered over a \(\mathbb{P}^3_\mathbb{C}\)-bundle over the dual torus \(\widehat T\) with fibers given by certain compactifications of Jacobians of curves of genus 5, and that \({\mathcal M}_3 \cong \text{Hilb}^6 (\widehat T) \times T\).
His primary tools are the Fourier-Mukai transform and Serre's construction. It turns out that certain extensions, constructed from zero-dimensional subschemes of \(T\), yield \(\mu\)-stable vector bundles if and only if those subschemes have the Cayley-Bacharach property. Also the boundary structures of the moduli spaces under consideration are described using geometric properties of certain zero-dimensional subschemes of \(T\). zero-dimensional subschemes of abelian variety; stable vector bundles; moduli space; compactification of Jacobians; Fourier-Mukai transform; Cayley-Bacharach property Maciocia, Zero-dimensional schemes on abelian surfaces, in: Zero-dimensional Schemes pp 253-- (1994) Algebraic moduli of abelian varieties, classification, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles Zero-dimensional schemes on abelian surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A real algebraic variety \(W\) of dimension \(m\) is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of \(\mathbb{R}^m\). Let \(l\) be any nonnegative integer. We prove that every map of class \(\mathcal{C}^l\) from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the \(\mathcal{C}^l\) topology by piecewise-regular maps of class \(\mathcal{C}^k\), where \(k\) is an arbitrary integer satisfying \(k \geq l\). Next we derive consequences regarding algebraization of topological vector bundles. real algebraic variety; piecewise-regular map; approximation; uniformly rational variety; piecewise-algebraic vector bundle Real algebraic sets, Real rational functions, Topology of vector bundles and fiber bundles, Semialgebraic sets and related spaces Approximation by piecewise-regular maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper gives a sufficient criterion for Schubert intersection numbers to vanish which can be executed in polynomial time. Schubert intersection numbers arise from Schubert varieties \(X_{w}\), which are certain subvarieties of the flag variety \(X\) indexed by permutations \(w \in S_{n}\). The Poincaré duals \(\sigma_{w} = [X_{w}]\) of Schubert varieties form a basis of the cohomology ring \(H^\ast(X)\) of the flag variety. A Schubert problem is a \(k\)-tuple \((w^{(1)}, w^{(2)}, \dots, w^{(k)})\) of permutations in \(S_{n}\) such that the sum of the lengths is \(\binom{n}{2}\). The Schubert intersection number \(C_{w^{(1)}, w^{(2)}, \dots, w^{(k)}}\) is equivalently either
\begin{itemize}
\item the multiplicity of \(\sigma_{w_{0}}\) in \(\prod_{i = 1}^k \sigma_{w^{(i)}}\), or
\item the number of points in \(\bigcap_{i = 1}^k g_{i}X_{w^{(i)}}\), where each \(g_{i}\) lies in some dense open subset of \(GL_{n}\).
\end{itemize}
Finding a combinatorial counting rule for Schubert intersection numbers is a famous open problem. Many algorithms exist for computing them. As stated previously, the aim of the paper is to give an algorithm to decide whether or not they vanish in a given case.
The algorithm which the paper introduces proceeds roughly as follows. Given \(w \in S_{n}\), one can define the Rothe diagram \(D(w)\) as a certain subset of the boxes of the \(n \times n\) grid. Given a Schubert problem \((w^{(1)}, w^{(2)}, \dots, w^{(k)})\), one concatenates the Rothe diagrams \(D(w^{(i)})\) to obtain a diagram \(D\). One then considers the fillings of \(D\) which obey certain rules. If there are no such fillings, then the Schubert intersection number vanishes. There is then a polynomial-time algorithm to determine whether there are indeed no such fillings.
The arguments of the paper use generalised permutahedra, which are obtained from the standard permutahedron by degeneration. The particular generalised permutahedra that the authors are interested in are known as ``Schubitopes''. These are constructed from fillings of rectangular grids. In the case of the Rothe diagram \(D(w)\) mentioned above, the Schubitope obtained is the Newton polytope of the Schubert polynomial \(\mathfrak{S}_{w}\) of \(w\) by [\textit{A. Fink} et al., Adv. Math. 332, 465--475 (2018; Zbl 1443.05179)]. The crux is that the emptiness of the set of fillings from the previous paragraph is equivalent to the presence of a certain point in a Schubitope by \textit{A. Adve} et al. [Sémin. Lothar. Comb. 82B, Paper No. 52, 12 p. (2019; Zbl 1436.05115)]. This can then be decided in polynomial time by standard linear programming methods, given the algorithm mentioned above.
The paper finishes by comparing the test for vanishing of Schubert intersection numbers proven in the paper to three other tests from the literature. In each instance, there are some cases in which the test from the paper can decide and which the test from the literature cannot, and also some cases where the opposite is true. We mention finally that the paper contains a couple of variations on its main result. Schubitopes; Newton polytopes; Schubert polynomials; Schubert intersection numbers Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Linear programming Generalized permutahedra and Schubert calculus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S \rightarrow C\) be a smooth projective surface with numerically trivial canonical bundle fibered onto a curve. We prove the multiplicativity of the perverse filtration with respect to the cup product on \(H^\ast(S^{[n]}, \mathbb{Q})\) for the natural morphism \(S^{[n]} \rightarrow C^{(n)}\). We also prove the multiplicativity for five families of Hitchin systems obtained in a similar way and compute the perverse numbers of the Hitchin moduli spaces. We show that for small values of \(n\) the perverse numbers match the predictions of the numerical version of the de Cataldo-Hausel-Migliorini \(P = W\) conjecture and of the conjecture by \textit{T. Hausel} et al. [Duke Math. J. 160, No. 2, 323--400 (2011; Zbl 1246.14063); Adv. Math. 234, 85--128 (2013; Zbl 1273.14101)]. Hitchin fibration; Hilbert scheme; perverse filtration Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Parametrization (Chow and Hilbert schemes) Multiplicativity of perverse filtration for Hilbert schemes of fibered surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Applying the multisection series method to the MacLaurin series expansion of arcsin-function, we transform the Apéry-like series involving the central binomial coefficients into systems of linear equations. By resolving the linear systems (for example, by \textit{Mathematica}), we establish numerous remarkable infinite series formulae for \(\pi\) and logarithm functions, including several recent results due to \textit{G. Almkvist} et al. [Exp. Math. 12, No. 4, 441--456 (2003; Zbl 1161.11419)] and \textit{D. Zheng} [Indian J. Pure Appl. Math. 39, No. 2, 137--155 (2008; Zbl 1170.11045)]. multisection series; Apéry-like series; central binomial coefficient Evaluation of number-theoretic constants, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Multisection method for Apéry-like series | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review, the authors consider a family of integral plane curves and prove that if the relative Hilbert scheme of points is smooth, then the pushforward of the constant sheaf on the relative Hilbert scheme contains no summands other than the expected ones.
As a corollary, they show that the perverse filtration on the cohomology of the compactified Jacobian of an integral plane curve encodes the cohomology of all Hilbert schemes of points on the curve. An interesting consequence for the enumerative geometry of Calabi-Yau three-folds is also mentioned. locally planar curves; Hilbert scheme; compactified Jacobian; versal deformation; perverse cohomology; decomposition theorem L. Migliorini and V. Shende, \textit{A support theorem for Hilbert schemes of planar curves}, J. Eur. Math. Soc. 15, 2353-2367. Parametrization (Chow and Hilbert schemes), Jacobians, Prym varieties, Formal methods and deformations in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) A support theorem for Hilbert schemes of planar curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The goal of this paper is to explore an abstract algebraic geometry background of multi-operator functional calculus. Results in this paper consisted of arguments from commutative algebra and algebraic geometry combined with the results of the joint spectral theory. First result provides a necessary condition when a functional calculus does exist for the module \(M\) over an open subset \(U\subseteq X\). After a spectrum of an algebraic variety over an algebraically closed field is introduced. Further, the author generalizes the concept of a spectrum introduced above to schemes. Noetherian schemes; quasi-coherent sheaf; spectrum of a module; sheaf cohomology Schemes and morphisms, Sheaves in algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Functional calculus in topological algebras, Homological dimension and commutative rings The spectrum of a module along scheme morphism and multi-operator functional calculus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a local Noetherian ring \((R,{\mathfrak m})\) and \(I\subset R\) an ideal of height \(>0\) the asymptotic depth \(\bar t\) of the higher conormal modules \(I^ n/I^{n+1}\) is related to some topology of the blowing up \(\pi\) of Spec(R) at I. One of the main results says that for large values of \(\bar t\) and \(\text{grad}e(I,R)>1\) the complement \(C:=\pi^{-1}(V(I))-\pi^{- 1}({\mathfrak m})\) of the special fiber in the exceptional fiber is highly connected (i.e. the connectedness subdimension of C is large, see proposition (4.8) for more details). The grade-condition in proposition (4.8) implies that the exceptional fiber is connected. Then the statement of (4.8) follows from proposition (3.7) where instead of the blowing-up morphism \(\pi\) one works with an arbitrary projective morphism induced by a homogeneous R-algebra S. In this more general context the author considers a Noetherian graded S-module \(M=\oplus M_ n\) and relates the depth of the R-modules \(M_ n\) to the connectivity of the sheaf \({\mathcal F}\) induced by M on Proj(S), see (3.3), (3.7) and (3.8). connectivity of sheafs; asymptotic depth; topology of the blowing up Brodmann, M, Asymptotic depth and connectedness in projective schemes, Proc. Am. Math. Soc., 108, 573-581, (1990) Dimension theory, depth, related commutative rings (catenary, etc.), Local rings and semilocal rings, Topological properties in algebraic geometry, Schemes and morphisms Asymptotic depth and connectedness in projective schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider design questions for an active optical lattice filter, which is being manufactured at the University of Texas at Dallas, and which has proven to be useful in the signal processing task of RF channelization. The filter can be described by a linear, discrete time state space model. The controlling agents, the gains, are embedded in the matrices intervening in this state space model. Consequently, techniques from linear feedback control theory do not apply. We concentrate on the question of finding real valued gains so that the \(A\) matrix of the state space model has a prescribed characteristic polynomial. We find that three of the coefficients can be arbitrarily picked, but that the remaining are constrained by these and the other system parameters. Our techniques use methods from constructive algebraic geometry. channelizers; photonics; Gröbner bases Signal theory (characterization, reconstruction, filtering, etc.), Filtering in stochastic control theory, Computational aspects in algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) Analysis of a polynomial system arising in the design of an optical lattice filter useful in channelization | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors consider the theory of gonality of a projective curve \({\mathcal C}\). The gonality of such a curve is the least integer \(d\) for which there exists a \(d:1\) morphism to the projective line. A gonal map is such a morphism of least degree. An algorithm is developed for determining such a gonal map for a given curve \({\mathcal C}\).
Such gonal maps are closely related to radical parametrizations of irreducible projective curves. An algorithm for determining the gonality of a curve can be used for deciding whether the gonality does not exceed 4. In this case the algorithm for computing a gonal map can be turned into an algorithm for computing a radical parametrization of the corresponding curve, as long as the characteristic of the coefficient field is different from 2 and 3. gonal map; canonical curves; scrolls; Betti table; syzygies; special linear series; radical parametrization Computational aspects of algebraic curves Computational aspects of gonal maps and radical parametrization of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{J. Riou} [J. Topol. 3, No. 2, 229--264 (2010; Zbl 1202.19004)] showed that operations on the higher \(K\)-theory of smooth schemes are completely determined by operations in degree zero. More precisely, if we denote by \(K\) a simplicial presheaf representing \(K\)-theory in the unstable motivic homotopy category of smooth schemes (over a fixed base), and if we view \(K_0\) as a presheaf on smooth schemes (over the same base), then there is a canonical bijection between operations \(K^n\to K\) and operations \(K_0^n\to K_0\).
In the present paper, the author extends this result in several directions. His main results extend Riou's theorem to (possibly singular) divisorial schemes by establishing natural bijections between the following sets of operations:
\begin{itemize}
\item operations \(K_0^n \to K_0\), where \(K_0\) is viewed as presheaf on the category of \emph{divisorial} schemes
\item operations \(K^n \to K\), where \(K\) is a representing simplicial presheaf in the unstable motivic homotopy category of smooth divisorial schemes
\item operations \(K^n \to K\), where \(K\) is viewed as an object in a homotopy category of simplicial presheaves on divisiorial schemes
\item operations \(K^n \to K\), where \(K\) is viewed as an object in a homotopy category of simplical presheaves on noetherian schemes
\end{itemize}
In the last two cases, the homotopy categories are defined with respect to Zariski local model structures.
While Riou assumes all schemes to be smooth and separated, the author, as a first step, observes that the arguments remain unaffected if we more generally consider smooth divisorial schemes. A key imput in the further generalizations are his previous results on embedding divisorial schemes into smooth ones [\textit{F. Zanchetta}, J. Algebra 552, 86--106 (2020; Zbl 1441.14005)].
The author moreover extends Riou's theorem to symplectic \(K\)-theory (the variant \(GW^{[2]}\) of hermitian \(K\)-theory in the notation of [\textit{M. Schlichting}, J. Pure Appl. Algebra 221, No. 7, 1729--1844 (2017; Zbl 1360.19008)]). The arguments in this case are very similar. \(K\)-theory; operations; divisorial schemes; motivic homotopy theory; hermitian \(K\)-theory; symplectic \(K\)-theory \(K\)-theory of schemes, Motivic cohomology; motivic homotopy theory, Homotopy theory Unstable operations on \(K\)-theory for singular schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems By using their technique of limit linear series, \textit{D. Eisenbud} and \textit{J. Harris} [Invent. Math. 87 (1987; 495--515, Zbl 0606.14014)] studied how Weierstrass points on general smooth curves degenerate when the smooth curve degenerates to a stable curve of compact type. In this article the situation is considered when the curve degenerates to a stable curve \(C_0\) of non-compact type which is a union of two irreducible smooth components meeting transversely at \(m>1\) points. The authors prove that if the components have genera \(\geq 1\) then the space parameterizing limits of Weierstrass schemes on \(C_0\) has dimension \(m-1\). Based on this theorem the limits of Weierstrass points on one component are given if one component has genus 1. Further details are deduced. For related results see also \textit{E. Esteves} and \textit{N. Medeiros} [C.R. Acad. Sci, Paris, Sér. I Math., 330, 873--878 (2000; Zbl 0972.14024)]. stable curves of non compact type; limits of Weierstrass points on reducible curves Coppens, Limit Weierstrass schemes on stable curves with 2 irreducible components, Atti Accad. Naz. Lincei 9 pp 205-- (2001) Riemann surfaces; Weierstrass points; gap sequences Limit Weierstrass schemes on stable curves with 2 irreducible components | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the last two decades, many computational problems arising in cryptography have been successfully reduced to various systems of polynomial equations. In this paper, we revisit a class of polynomial systems introduced by Faugère, Perret, Petit and Renault [\textit{J.-C. Faugère} et al., Eurocrypt 2012, Lect. Notes Comput. Sci. 7237, 27--44 (2012; Zbl 1290.94070)]. Based on new experimental results and heuristic evidence, we conjecture that their degrees of regularity are only slightly larger than the original degrees of the equations, resulting in a very low complexity compared to generic systems. We then revisit the application of these systems to the elliptic curve discrete logarithm problem (ECDLP) for binary curves. Our heuristic analysis suggests that an index calculus variant due to Diem requires a subexponential number of bit operations \(O(2^{c\cdot n^{\frac{2}{3}}\cdot \log n})\) over the binary field \(\mathbb{F}_{2^n}\), where \(c\) is a constant smaller than~2. According to our estimations, generic discrete logarithm methods are outperformed for any \(n > N\) where \(N \approx 2000\), but elliptic curves of currently recommended key sizes \((n \approx 160)\) are not immediately threatened. The analysis can be easily generalized to other extension fields. Petit, C., Quisquater, J.-J.: On polynomial systems arising from a Weil descent. In: ASIACRYPT 2012, pp. 451--466 (2012). Citations in this document: {\S}1.3 Cryptography, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry On polynomial systems arising from a Weil descent | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A system of s distinct lines \(L_ 1,...L_ s\), passing through the origin in the affine space \({\mathbb{A}}_ k^{n+1}\) (k denoting an algebraically closed field), defines a simple curve singularity. The problem of computing the CM type \(t(L_ 1,...L_ s)\) of the local ring at the origin was considered by \textit{A. V. Geramita} and \textit{F. Orecchia} [J. Algebra 70, 116-140 (1981; Zbl 0464.14007)], who mostly dwelled on the case when the ''direction numbers'' \(P_ 1,...P_ s\) of those lines, considered as points of \({\mathbb{P}}^ n\), are in generic s- position. By definition, this means that the \(s\times (_ n^{d+n})\) matrix G(d), obtained by evaluating all degree d monomials on \(n+1\) indeterminates at the homogeneous coordinates of \(P_ 1,...,P_ s\), has maximal possible rank for every \(d\geq 1.-\) The present authors obtain a general formula in the form: \(t(L_ 1,...L_ s)=\sum^{s- 1}_{j=r}\gamma (j)+rk G(s-1)-(_{\quad n}^{n+s-2})\) where the \(\gamma\) (j)'s are - in principle - computable from certain matrices obtained from G by row-transformations (one has also to assume the leading entries of the direction numbers equal to 1, but this is no restriction on the generality); the integer r is the minimal degree of s hypersurface passing through all s points; \(s\geq 2\) and \(n\geq 2.\)
This is used to deduce a useful expression in the case of generic s- position: \(t(L_ 1,...,L_ s)=\gamma (r)+s-(_{\quad n}^{r+n-1}).\) Examples are given to show that for s and n fixed, the type can vary even if the lines are in generic s-position. lines in affine space; Cohen-Macaulay ring; Gorenstein ring; singularity; lines in generic position; CM type; local ring at the origin Baruch, M.; Brown, W. C.: A matrix computation for the Cohen-Macaulay type of s-lines in affine (n + 1) -space. J. algebra 85, 1-13 (1983) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) A matrix computation for the Cohen-Macaulay type of s-lines in affine \((n+1)\)-space | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems R. Vakil defined singularity type as an equivalence class of pointed schemes under the relation generated by \((X,x)\sim(Y,y)\) if there is a smooth morphism \((X,x)\rightarrow(Y,y)\) [\textit{R. Vakil}, Invent. Math. 164, No. 3, 569--590 (2006; Zbl 1095.14006)]. He showed that every singularity type over \(\mathbb Z\) appears on various moduli spaces of well-behaved objects: one says that Murphy's law holds for these moduli spaces.
The author investigates the Hilbert scheme of points, which is missing in Vakil's list. The main theorem states that Murphy's law holds up to retraction for \(\mathrm{Hilb}_{\mathrm{pts}}(\mathbb A^{16}_{\mathbb Z})\). The proof proceeds by a series of reductions from objects with more structure. The main role is played by a generalized Białynicki-Birula decomposition [\textit{J. Jelisiejew} and \textit{Ł. Sienkiewicz}, J. Math. Pures Appl. (9) 131, 290--325 (2019; Zbl 1446.14030)]. In order to construct the local retractions, the author uses TNT frames.
Using concrete singularity types, the author shows that \(\mathrm{Hilb}_{\mathrm{pts}}(\mathbb A^{16}_{\mathbb Z})\) and \(\mathrm{Hilb}_{\mathrm{pts}}(\mathbb A^{16}_{\mathbb C})\) are non-reduced, answering questions raised by J. Fogarty [\textit{J. Fogarty}, Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. He also shows that not all finite schemes over finite fields lift to characteristic zero, answering a question by R. Hartshorne [\textit{R. Hartshorne}, Deformation theory. Berlin: Springer (2010; Zbl 1186.14004)].
As a corollary of the main theorem, Murphy's law holds up to retraction for \(\mathrm{Hilb}_{\mathrm{pts}}(\mathbb P^{16}_{\mathbb Z})\). Since the forgetful functor from embedded to abstract deformations of a finite scheme is smooth, the above described pathologies appear for abstract deformations of finite schemes.
The author highlights that the choice of ambient dimension \(n=16\) was made for sake of transparency in the proof, but he suggests that the result may hold for \(n=6\) or even \(n=4\). Hilbert scheme of points; Vakil's Murphy's Law Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Local deformation theory, Artin approximation, etc., Deformations and infinitesimal methods in commutative ring theory, Deformations of singularities Pathologies on the Hilbert scheme of points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems No review copy delivered. Cryptography, Applications to coding theory and cryptography of arithmetic geometry, Data encryption (aspects in computer science) An RNS implementation of an \( \mathbb F_p\) elliptic curve point multiplier | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a variety over a valued field, one can tropicalize it and construct a combinatorial object. Originally, these tropical varieties are polyhedral complexes which inherit topology from \(\mathbb{R}^n\). \textit{J. Giansiracusa} and \textit{N. Giansiracusa} [Duke Math. J. 165, No. 18, 3379--3433 (2016; Zbl 1409.14100)] combined \(\mathbb{F}_1\)-geometry with the notion of bend loci to equip tropical varieties with scheme structure, and therefore, obtained tropical schemes. This paper investigates Picard groups of tropical schemes as a first step towards building scheme-theoretic tropical divisor theory which can lead to finding a scheme-theoretic tropical Riemann-Roch theorem.
For monoid \(M\) one can pass from monoid scheme \(X = \text{Spec}\, M\) to scheme \(X_K = \text{Spec}\, K[M]\) by scalar extension to field \(K\). \textit{J. Flores} and \textit{C. Weibel} [J. Algebra 415, 247--263 (2014; Zbl 1314.14003)] show Picard groups \(\text{Pic} (X)\) and \(\text{Pic} (X_K)\) are isomorphic. The current paper proves this isomorphism in tropical setting: for irreducible monoid scheme \(X\) and idempotent semifield \(S\) Picard groups \(\text{Pic} (X)\) and \(\text{Pic} (X_S)\) are both isomorphic to certain sheaf cohomology groups, and hence, are isomorphic. They also construct the group \(\text{CaCl}\, (X_S)\) of Cartier divisors modulo principal Cartier divisors and show that \(\text{CaCl}\, (X_S)\) is isomorphic to \(\text{Pic} (X_S)\). tropical schemes; Picard groups; Cartier divisors; idempotent semiring Geometric aspects of tropical varieties, Picard groups, Semirings, Semifields, Ordered semigroups and monoids Picard groups for tropical toric schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give an explicit algorithm computing the motivic generalization of the Poincaré series of a plane curve singularity introduced by \textit{A. Campillo} et al. [Monatsh. Math. 150, No. 3, 193--209 (2007; Zbl 1111.14020)]. It is done in terms of the embedded resolution. The result is a rational function depending of the parameter \(q\), at \(q=1\) it coincides with the Alexander polynomial of the corresponding link. For irreducible curves we relate this invariant to the Heegaard-Floer knot homology constructed by P. Ozsváth and Z. Szabó. Many explicit examples are considered. E. Gorsky, Combinatorial computation of the motivic Poincaré series, Journal of Singularities 3 (2011) 48 [ arXiv:0807.0491 ]. Invariants of knots and \(3\)-manifolds, Singularities of curves, local rings, Arcs and motivic integration, Milnor fibration; relations with knot theory, Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms Combinatorial computation of the motivic Poincaré series | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Gysin map introduced by W. Gysin, originally associated to a map $f:M\rightarrow N$ of closed oriented manifolds, a push-forward covariant homomorphism $f_*: H^*(M)\rightarrow H^*(N)$ between cohomology groups and was later generalized to many different settings. The Gysin homomorphism for fiber bundles and de Rham cohomology of differential forms has a natural interpretation as integration along fibers. The Gysin map was also defined as a push-forward in Chow groups of varieties along proper morphisms of non-singular algebraic varieties.
Gysin maps proved useful in singularity theory and have provided a tool to study the degeneracy loci of morphisms of flag bundles which are related to Schubert manifolds by the Thom-Porteous formula. Most of the results on push-forwards for flag bundles relied on inductive procedures reducing the problem to studying projective bundles. The study of the degeneracy loci of morphisms of flag bundles leads to the development of combinatorial techniques concentrating on the study of Schur polynomials and their generalizations and modifications.
Another direction of study of Gysin homomorphisms was motivared by Quillen's description of the push-forward maps in complex cobordism using a certain type of a residue, which provided a background for the results of Damon and Akyildiz and Carrell expressing Gysin maps for fiber bundles as Grothendieck residues.
The development of equivariant cohomology had enriched the theory with new tools and a different perspective. Many classical theorems have been rephrased in terms of equivariant characteristic classes. The question of computing Gysin maps for projective bundles can be reduced to studying push-forward maps in equivariant cohomology. A powerful tool to study Gysin maps in equivariant cohomology are localization theorems.
In this paper the author gives a review and an example of computational application of an adaptation in the context of equivariant cohomology of the Pragacz-Ratajski formula for push-forwards of Schur classes for Lagrangian Grassmaniann manifolds. Gysin maps; push-forward maps; characteristic classes; equivariant cohomology; Schubert manifolds; flag manifolds; localization; Grothendieck residue; Lagrangian Grassmanianns Equivariant algebraic topology of manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Pushing-forward Schur classes using iterated residues at infinity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging from sums of squares to the structure of absolute Galois groups. Here, we survey some recent work on generalizations of the Milnor conjecture to the context of schemes (mostly smooth varieties over fields of characteristic not 2). Surprisingly, a version of the Milnor conjecture fails to hold for certain smooth complete \(p\)-adic curves with no rational theta characteristic (this is the work of Parimala, Scharlau, and Sridharan). We explain how these examples fit into the larger context of an unramified Milnor question, offer a new approach to the question, and discuss new results in the case of curves over local fields and surfaces over finite fields. Milnor conjecture; smooth complete \(p\)-adic curves Auel, A.: Remarks on the Milnor conjecture over schemes, Galois-Teich üller theory and arithmetic geometry. Adv. Stud. Pure Math. \textbf{63}, 1-30 (2012) Research exposition (monographs, survey articles) pertaining to \(K\)-theory, Research exposition (monographs, survey articles) pertaining to number theory, Quadratic forms over general fields, Algebraic theory of quadratic forms; Witt groups and rings, Witt groups of rings, Brauer groups of schemes, Brauer groups (algebraic aspects), Higher symbols, Milnor \(K\)-theory Remarks on the Milnor conjecture over schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(P_{1},\dots, P_{s}\) be a finite set of \(s\) distinct points in a projective space and let \(m_{1}, \dots, m_{s}\) be positive integers. Consider homogeneous polynomials that vanish at \(P_{i}\) to order \(m_{i}\) for all \(i \in \{1,\dots, s\}\). The set of all such polynomials is the homogeneous ideal \(I_{X}\) of the fat point scheme \(X = \sum_{i=1}^{s} m_{i}P_{i}\). The vector space dimension of the degree \(d\) polynomials in \(I_{X}\) is know if \(d\) is large enough. In geometric terms, we can say that the fat point scheme \(X\) imposes independent conditions on forms of degree \(d>>0\). The least integer \(d\) such that this is true for degree \(d\) forms is called the regularity index \(X\), denoted here by \(r(X)\). It was conjectured that \(r(X) \leq \mathrm{Seg}(X)\), where
\[\mathrm{Seg}(X) := \max\bigg\{ \frac{-1 + \sum_{P_{i} \in L}m_{i}}{\dim L} \, : \, L \subset \mathbb{P}^{n}\text{ is a positive dimensional linear subspace}\bigg\}.\]
The number \(\mathrm{Seg}(X)\) is called the Segre bound. In this extremely interesting paper the authors show that for every fat point scheme \(X\) of some projective space one always has \(r(X) \leq\mathrm{Seg}(X)\). The proof of the authors is based on a new partition result for matroids (Proposition 2.6 therein) that might be of independent interest. fat point schemes; regularity Divisors, linear systems, invertible sheaves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Configurations and arrangements of linear subspaces, Syzygies, resolutions, complexes and commutative rings, Combinatorial aspects of matroids and geometric lattices Segre's regularity bound for fat point schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper studies n-dimensional varieties in \({\mathbb{P}}^ N\) which are hypersurfaces in some linear subspace. These are called n-hypersurfaces. The first theorem in the paper states that if \(n>0\), then a subscheme of \({\mathbb{P}}^ N\) with the Hilbert polynomial of an n-hypersurface is indeed an n-hypersurface. It follows that such a Hilbert polynomial is minimal among all Hilbert polynomials of subschemes of the same dimension and degree.
The second result is the following: Let X be a projective scheme with Hilbert polynomial \(P(t)=\sum^{n}_{i=0}a_ i\left( \begin{matrix} t+n-i\\ n-i\end{matrix} \right).\) Then \(2a_ 1\geq a_ 0(a_ 0-1)\), and if X is of pure dimension, equality implies that X is an n-hypersurface (of degree \(a_ 0)\). Hilbert scheme; n-hypersurfaces; Hilbert polynomial Ådlandsvik, B.: Hilbert schemes of hypersurfaces and numerical criterions. Math. Scand.56, 163-170 (1985) Parametrization (Chow and Hilbert schemes), \(n\)-folds (\(n>4\)), Fine and coarse moduli spaces, Special surfaces Hilbert schemes of hypersurfaces and numerical criterions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the relations between Chow groups of \(0\)-cycles and the cohomology of the Milnor \(K\)-theory sheaf. The main results (Theorems 1.1 and 1.2) are generalizations of Bloch's formula for smooth schemes: if \(k\) is a perfect field and \(X\) is a reduced quasi-projective \(k\)-scheme of dimension \(d\), and denote by \(\mathcal{K}^M_d\) the Milnor \(K\)-theory Nisnevich sheaf (Definition 2.5), then there is an isomorphism
\[
\mathrm{CH}_0(X)\simeq H^d_{Nis}(X,\mathcal{K}^M_d)
\]
in the following cases: 1) if we take \(\mathrm{CH}_0(X)\) to be the Levine-Weibel Chow group of \(0\)-cycles (\S 2.1), \(k\) is algebraically closed, and \(X\) either affine, or projective and regular in codimension \(1\); 2) if we take \(\mathrm{CH}_0(X)\) to be a variant of the usual Chow group of \(0\)-cycles (\S 2.1) and \(X\) is an affine surface. From these results the authors deduce several variants: for Chow groups with modulus (Theorems 1.3, 1.4 and 1.9), for motivic cohomology with modulus (Theorem 1.6 and 1.7), and a theorem of Schlichting on splittings of vector bundles over affine schemes (Corollary 1.10). Chow group of 0-cycles; Chow groups with modulus; Milnor \(K\)-theory Algebraic cycles, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) \(K\)-theory and 0-cycles on schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper introduces formally the concept of local factorizability used in earlier factorizability work, identifies the basic form of obstructions to local factorizability of birational morphisms, and outlines a combinatorial technique for analyzing such obstructions. As an application and illustration, two open cases in the classification of birational morphisms with small canonical divisors are settled. threefolds; local factorizability of birational morphisms M. Schaps,Combinatorial analysis of point obstructions to local factorizability in three folds, preprint, Bar Ilan University. Rational and birational maps, \(3\)-folds, Birational automorphisms, Cremona group and generalizations Combinatorial analysis of point obstructions to local factorizability in three-folds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a 0-dimensional scheme \(X\) which is contained in an irreducible curve \(C\) in \(\mathbb{P}^3\), which in turn is contained in a smooth quadric surface \(Q\). Which are the possibilities for the Hilbert function (postulation) of such an \(X\)?
The paper gives an answer (i.e a complete characterization of all possible Hilbert functions for which it exists \(X\) as above) when \(C\) is ``good enough'', in particular when \(C\) is a complete intersection, arithmetically Cohen-Macaulay, or arithmetically Buchsbaum.
In the general case conditions on the Hilbert function are given. 0-dimensional schemes; postulation; quadric surface; Hilbert function Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Plane and space curves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Postulation of subschemes of irreducible curves on a quadric surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the present work, we extend the standard idea of numerical parameterization (i.e., parameterization by the numerical solution of initial-value problems (IVPs) for ordinary differential equations (ODEs)) to affine varieties in \(\mathbb C^n\) for \(n \geq 2\). We use these results with an efficient implementation in Maple to explore the use of numerical parameterization for the visualization of Riemann surfaces. symbolic-numeric algorithms; numerical polynomial algebra Aruliah, D.A., Corless, R.M.L: Numerical parameterization of affine varieties using ODEs. In: Gutierrez, J. (ed.) Proc. ISSAC, pp. 12--18. ACM Press, New York (2004) Symbolic computation and algebraic computation, Numerical methods for ordinary differential equations, Computational aspects of higher-dimensional varieties Numerical parameterization of affine varieties using ODEs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F\) be a finite field, \(K\) a quadratic extension of \(F\), \(\sigma\) the unique involution of \(K\mid F\); extend \(\sigma\) to an involution of the \(F\)-algebra \(A=K[X_0, \dots, X_n,Y_0,\dots,Y_n]\) by posing \(\sigma(X_i) =Y_i\), \(\sigma(Y_i) =X_i\) for \(0\leq i\leq n\). A bihomogeneous form \(f\in A\) of bidegree \((d,d)\) is called Hermitian if \(\sigma (f)=f\); the set \(H\) of all Hermitian forms is a finite-dimensional vector \(F\)-space. Denoting by \(W\) the associated projective space, there is a natural map \(h:\mathbb{P}_n(K)\to W(F)\). The authors determine a ``large'' constant \(c\) (depending on \(n\), \(d\) and \(\text{Card}\,F)\) such that any \(c\) distinct points in \(\mathbb{P}_n(K)\) are sent by \(h\) to linearly independent points in \(W(F)\). finite field; Hermitian forms; Hermitian Veronesean variety; bihomogeneous polynomial; Segre embedding Finite ground fields in algebraic geometry, Schemes and morphisms, Combinatorial aspects of finite geometries Hermitian Veronesean schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Two proper polynomial mappings \(f_{1},f_{2}:\mathbb{C}^{n}\rightarrow \mathbb{C}^{n}\) are said to be equivalent if there exist polynomial automorphisms \(\Phi _{1},\Phi _{2}\) of \(\mathbb{C}^{n}\) such that \( f_{2}=\Phi _{2}\circ f_{1}\circ \Phi _{1}.\) The authors describe some of their recent and new results concerning this equivalence. In particular they give:
1. Discrete and continuous families of proper polynomial mappings whose generic members are not pairwise equivalent (for \(n=2\) and topological degree \( d\geq 3\); for \(d=2\) there is only one equivalence class represented by the mapping \((x,y)\mapsto (x,y^{2})\)).
2. The complete classification of the equivalence classes for mappings \(f: \mathbb{C}^{2}\rightarrow \mathbb{C}^{2}\) which are Galois coverings (i.e. being the Galois covering outside the branch locus of \(f\)) with finite Galois group.
3. Some partial results concerning the same topics for \(n\geq 3.\) proper polynomial mapping; Galois covering Bisi, C.; Polizzi, F.: Proper polynomial self-maps of the affine space: state of the art and new results. Contemp. math. 553, 15-25 (2011) Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Rational and birational maps Proper polynomial self-maps of the affine space: state of the art and new results | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We survey some recent development of the theory of local models for Shimura varieties. local models; Shimura varieties; nearby cycles; affine flag varieties Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties A survey of local models of Shimura varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We define the curvilinear rank of a degree \(d\) form \(P\) in \(n+1\) variables as the minimum length of a curvilinear scheme, contained in the \(d\)-th Veronese embedding of \(\mathbb{P}^n\), whose span contains the projective class of \(P\). Then, we give a bound for rank of any homogenous polynomial, in dependance on its curvilinear rank. maximum rank; curvilinear rank; curvilinear schemes; cactus rank , Curvilinear schemes and maximum rank of forms, Matematiche (Catania) 72 (2017), no. 1, 137--144. Projective techniques in algebraic geometry Curvilinear schemes and maximum rank of forms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The articles of this volume will be reviewed individually.
Publisher's description: Robin Hartshorne's classical 1966 book ''Residues and Duality'' [RD] developed Alexandre Grothendieck's ideas for a pseudofunctorial variance theory of residual complexes and duality for maps of noetherian schemes.
The three articles in this volume rework the main parts of the last two chapters in [RD], in greater generality--for Cousin complexes on formal schemes, not just residual complexes on ordinary schemes--and by more concrete local methods which clarify the relation between local properties of residues and global properties of dualizing pseudofunctors. A new approach to pasting pseudofunctors is applied in using residual complexes to construct a dualizing pseudofunctor over a fairly general category of formal schemes, where compactifications of maps may not be available.
A theory of traces and duality with respect to pseudo-proper maps is then developed for Cousin complexes. For composites of compactifiable maps of formal schemes, this, together with the above pasting technique, enables integration of the variance theory for Cousin complexes with the very different approach to duality initiated by Deligne in the appendix to [RD].
The book is suitable for advanced graduate students and researchers in algebraic geometry.
Table of contents: J. Lipman, S. Nayak, and P. Sastry, Part 1. Pseudofunctorial behavior of Cousin complexes on formal schemes (3--133); P. Sastry, Part 2. Duality for Cousin complexes (137-192); S. Nayak, Part 3. Pasting pseudofunctors (195--271); Index (273--276) Lipman, J., Nayak, S., Sastry P.: Variance and duality for Cousin complexes on formal schemes. In: Pseudofunctorial Behavior of Cousin Complexes on Formal Schemes. Contemp. Math., vol. 375, pp. 3--133. American Mathematical Society, Providence (2005) Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Collections of articles of miscellaneous specific interest Variance and duality for Cousin complexes on formal schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We associate with \(n+1\) different points in \(\mathbb{P}^1(\mathbb{C})\) a system of nonlinear differential equations that gives for \(n=1\) the KP-hierarchy. To this geometric configuration there corresponds naturally a basic space \(H^*\) and a group of commuting flows \(\widetilde\Gamma\). Starting from the Grassmann manifold corresponding to \(H^*\), we construct wavefunctions that yield solutions of the differential equations in the derivatives with respect to the flow parameters. KP-hierarchy; Grassmann manifold; wavefunctions KdV equations (Korteweg-de Vries equations), Grassmannians, Schubert varieties, flag manifolds, Meromorphic functions of one complex variable (general theory) A multipoint version of the KP-hierarchy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A degeneration $X\rightarrow C$ gives a degeneration of the corresponding Hilbert schemes, and the study of such degenerations is of obvious interest in algebraic geometry. \par New techniques for studying degenerations were introduced by \textit{J. Li} and \textit{B. Wu} [Commun. Anal. Geom. 23, No. 4, 841--921 (2015; Zbl 1349.14014)], with an approach based on the technique of expanded degenerations. The method is very general and can be used to study degenerations of various types of moduli problems such as Hilbert schemes and moduli spaces of sheaves. They also used degenerations of Quot-schemes and coherent systems to obtain degeneration formulae for Donaldson-Thomas invariants and Pandharipande-Thomas stable pairs. \par In the present article, the authors search to understand degenerations of irreducible holomorphic symplectic manifolds. As a start, the authors study degenerations of $K3$ surfaces and their Hilbert schemes, a guiding example being type II degenerations of $K3$ surfaces leading to the investigation of degenerations of the Hilbert scheme of points for simple degenerations $X\rightarrow C$ with no a priori restriction on the type or dimension of the fibre. The assumption of a simple degeneration gives that the total space is smooth, and that the central fibre $X_0$ over the point $0\in C$ of the $1$-dimensional base $C$ has normal crossing along smooth varieties. In this paper, a technique for the construction of degenerations of Hilbert schemes is developed, which allow to control the geometry of the degenerate fibres. \par The authors describe the equalities and differences between their approach and that of Li and Wu [loc. cit.]: First of all, this article considers only Hilbert schemes of points, whereas Li and Wu consider general Hilbert schemes of ideal sheaves with arbitrary Hilbert polynomial, and also Quot schemes. Both approaches apply Li's method of expanded degenerations $X[n]\rightarrow C[n]$ which in the case of constant Hilbert polynomial means the construction of a family whose special fibre over $0$ parametrises length $n$ subschemes of the degenerate fibre $X_0.$ The main problem in this setting is to describe the subschemes whose support meets the singular locus of $X_0,$ and the idea of the construction is that whenever a subscheme approaches a singularity in $X_0,$ a new ruled component is inserted into $X_0$ and it will be sufficient to work with subschemes supported on the smooth loci of the fibres of $X[n]\rightarrow C[n].$ The dimension of the base $C[n]$ is increased at each step of increasing $n,$ and finally one has to take equivalence classes of subschemes supported on the fibres of $X[n]\rightarrow C[n].$ Additionally, the construction of expanded degenerations also includes an action of an $n$-dimensional torus $G[n]\subset\text{SL}(n)$ acting on $X[n]\rightarrow C[n]$ such that $C[n]/\!\!/ G[n]=C.$ \par Li and Wu proceed by constructing the stack $\mathfrak{X}/\mathfrak{C}$ of expanded degenerations associated to $X\rightarrow C,$ giving a notion of equivalence. For fixed Hilbert polynomial $P,$ a definition of stable ideal sheaf is given, and this is used to define a stack $I^P_{\mathfrak{X/C}}$ over $C$ parametrizing those (the Li-Wu stack). In the case where the constant Hilbert polynomial $P=n,$ this gives subschemes of length $n$ supported on the smooth locus of a fibre of an expanded degeneration, having finite automorphism group. In the present article, the method is not to use the Li-Wu stack, but to use GIT with respect to the action of $G.$ \par The body of the article is the construction of a setup allowing to apply GIT-methods. One has to assume that the dual graph $\Gamma(X_0)$ associated to the singular fibre $X_0$ is bipartite, i.e. that it has no cycles of odd length. One can always perform a quadratic base change so that this holds, so the assumption is rather mild. At first, the authors construct a relatively ample line bundle $\mathcal L$ on $X[n]\rightarrow C[n].$ Then the bipartite assumption allows to construct a $G[n]$-linearisation on $\mathcal L$ which is proven to be applicable to Hilbert schemes. The choice of the correct $G[n]$- linearisation is the most important technical tool of the present article. Using $\mathcal L$, the authors construct an ample line bundle $\mathcal M_l$ on the relative Hilbert scheme defined by $\mathbf{H}^n:=\text{Hilb}^n(X[n]/C[n]),$ with a natural $G[n]$-linearisation. In this explicit situation, GIT stability can be analysed using a relative version of the Hilbert-Mumford numerical criterion: It is proved that (semi-)stability of a point $[Z]\subset\mathbf{H}^n$ only depends on the degree $n$-cycle associated to $Z.$ \par Fixing the $G[n]$-linearised sheaf $\mathcal L,$ the construction depends on several choices: The orientation of the dual graph $\Gamma(X_0)$ has two possible choices as it is bipartite, and both give isomorphic GIT quotients. A suitable $l$ is chosen in the construction of $\mathcal M_l,$ but the characterisation of stable $n$-cycles shows that the final result is independent of this. \par Let $Z\subset X[n]_q$ for some point $q\in C[n].$ Using a local étale coordinate $t$ it is obtained coordinates $t_1,\dots,t_{n+1}$ on $C[n]$ and then $\{a_1,\dots,a_r\}$ is defined to be the subset indexing coordinates with $t_i(q)=0.$ Put $a_0=1,\;a_{r+1}=n+1,$ then $\mathbf{a}=(a_0,\dots,a_{r+1})\in\mathbb Z^{r+2}$ determines a vector $\mathbf{v_a}\in\mathbb Z^{r+1}$ with $i$-th component $a_i-a_{i-1}.$ Now, by definition, $Z$ has smooth support if it is supported in the smooth part of the fibre $X[n]_q.$ Then each point $P_i$ in the support of $Z$ is contained in a unique component of $X[n]_q$ with multiplicity say $n_i.$ This leads to the definition of the numerical support $\mathbf{v}(Z)\in\mathbb Z^{r+1}.$ \par The first main result then describes the stable locus in $\mathbf{H}^n$ with respect to $\mathcal M_l$ when $l\gg 2n^2:$ If $[Z]\in\mathbf{H}^n$ has smooth support, then $[Z]\in\mathbf{H}^n(\mathcal M_l)^{\text{ss}}$ if and only if $\mathbf{v}(Z)=\mathbf{v_a}$ (then also $[Z]\in\mathbf{H}^n_{\text{GIT}}:=\mathbf{H}^n(\mathcal M_l)^s$), if $[Z]\in\mathbf{H}^n$ does not have smooth support, then $[Z]\notin\mathbf{H}^n(\mathcal M_l)^{\text{ss}}.$ \par Given this result, the authors define the main object of study in the paper, the GIT quotient $I_{X/C}=\mathbf{H}_{\text{GIT}}^n/G[n].$ The advantage is that the GIT stable points can be controlled very explicitly so that the geometry of the fibres of the degenerate Hilbert-schemes can be controlled in detail. On the other side, the authors define the stack quotient $\mathcal I_{X/C}=[\mathbf{H}^n_{\text{GIT}}/G[n]].$ The next main result gives the relation between the two concepts, saying that the GIT quotient $I^n_{C/X}$ is projective over $C,$ and that the stack $\mathcal I^n_{X/C}$ is a Deligne-Mumford stack, proper and of finite type over $C$ with $I_{X/C}$ as coarse moduli space. In addition, the morphism $f:\mathcal I^n_{X/C}\rightarrow T^n_{\mathfrak{X/C}}$ is an isomorphism of Deligne-Mumford stacks. \par The above results prove that the GIT approach and the Li-Wu construction of degenerations of Hilbert schemes of points are equivalent. One advantage are the tools to explicitly describe the degenerate Hilbert schemes, and this is thoroughly illustrated with an example of degree $n$ Hilbert schemes on two components. \par As mentioned, a main objective of the article is to construct good degenerations of Hilbert schemes of $K3$ surfaces. It turns out that the technique with relative Hilbert schemes cannot be applied directly, but the situation is analysed based on the results in the paper, and a sketch of further work is given. \par This is a very deep and good article, showing the force of GIT theory compared to stack-theory. This proves that techniques of classical algebraic geometry are appropriate for solving algebraic geometric problems, and gives the theory for further studies (in particular of degenerations of Hilbert schemes of $K3$ surfaces). In addition, the article is very well written, and can be used as a guide for authors in the field. GIT; geometric invariant theory; degeneration; Hilbert scheme; expanded degenerations; Donaldson-Thomas invariants; Pandharipande-Thomas stable pairs; $K3$ surfaces; Hilbert scheme of points; Li-Wu stack; bipartite graph; smooth support; numerical support; GIT quotient; stack quotient; Deligne-Mumford stack Fibrations, degenerations in algebraic geometry, Parametrization (Chow and Hilbert schemes), Geometric invariant theory, Stacks and moduli problems A GIT construction of degenerations of Hilbert schemes of points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We construct a substitute for principal parts for systems of algebras with a natural local presentation by free modules. The main application of the theory is to families of locally complete intersection curves. In that case we obtain the sheaves of \textit{E. Esteves} [Ann. Sci. École Norm. Supér. 29, 107--134 (1996; Zbl 0872.14025)]. The advantage of the greater generality is that it makes the theory much more transparent. It also makes it possible to avoid many technical difficulties. Weierstrass point; Gorenstein curves; algebra of jets; principal parts Laksov D., Special issue in honor of Steven L. Kleiman, Comm. in Algebra 31 pp 4007-- (2003) Families, moduli of curves (algebraic), Regular local rings, Riemann surfaces; Weierstrass points; gap sequences Wronski systems for families of local complete intersection curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a group scheme. The notion of a diagram of schemes was introduced by [\textit{M. Hashimoto}, Foundations of Grothendieck duality for diagrams of schemes. Lecture Notes in Mathematics 1960. Berlin: Springer. (2009; Zbl 1163.14001)]. His purpose is to study \(G\)-linearized quasi-coherent sheaves over \(G\)-schemes and \((G, A)\)-modules where \(A\) is a \(G\)-algebra.
In the present paper, the authors established the theory of local cohomology for diagrams of schemes. As a consequence, they generalized the theorem of \textit{M. Hochster} and \textit{J. A. Eagon} [Am. J. Math. 93, 1020--1058 (1971; Zbl 0244.13012)]. diagram of schemes; group scheme; invariant theory; local cohomology DOI: 10.1307/mmj/1220879415 Generalizations (algebraic spaces, stacks), Actions of groups on commutative rings; invariant theory, Group actions on varieties or schemes (quotients) Local cohomology on diagrams of schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be an irreducible smooth complex projective curve of genus \(g\geq 2\) and let \(C_d\) be its \(d\)-fold symmetric product. In this paper, we study the question of semi-orthogonal decompositions of the derived category of \(C_d\). This entails investigations of the canonical system on \(C_d\), in particular its base locus. semi-orthogonal decomposition; symmetric product; gonality; Albanese map Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Jacobians, Prym varieties, Special divisors on curves (gonality, Brill-Noether theory) Semi-orthogonal decomposition of symmetric products of curves and canonical system | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We introduce different model structures on the categories of simplicial presheaves and simplicial sheaves on some essentially small Grothendieck site \(T\) and give some applications of these simplified model categories. In particular, we prove that the stable homotopy categories \(\text{SH}((\text{Sm}/k)_{\text{Nis}},\mathbb{A}^1)\) and \(\text{SH}((\text{Sch}/k)_{\text{cdh}},\mathbb{A}^1)\) are equivalent. This result was first proved by Voevodsky and our proof uses many of his techniques, but it does not use his theory of Delta-closed classes. Blander, B. A., \textit{local projective model structures on simplicial presheaves}, J. K-Theory, 24, 283-301, (2001) Homotopy theory and fundamental groups in algebraic geometry, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Simplicial sets, simplicial objects (in a category) [See also 55U10], Topological categories, foundations of homotopy theory Local projective model structures on simplicial presheaves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(W\) be the big Witt vector scheme over a reduced base ring \(R\). The authors provide a \(q\)-deformation of \(W\) to a non-unital ring scheme \(W^{(q)}\) which is enhanced with a Verschiebung as well as a Frobenius lift. This deformation is isomorphic to the \(q\)-deformation provided in [\textit{Y.-T. Oh}, J. Algebra 309, No. 2, 683--710 (2007; Zbl 1119.13018)], but the description provided here is simpler.
In detail, let \(S\) be a divisor stable set of natural numbers (so if \(n\in S\) then each positive integer divisor of \(n\) is in \(S\)). Then \(W_S(A)\) is the usual ring of \(S\)-Witt vectors with coefficients in a commutative ring \(A\). The authors define \(W_S^{(q)}(A)=W_S(A^{(q)})\), where \(A\) is a unital algebra over \(\mathbb{Z}[q]\) and \(A^{(q)}=A\) as additive groups with twisted multiplication \(a\ast b = qab\). Witt ring; \(q\)-deformation Witt vectors and related rings, Group schemes The universal deformation of the Witt ring scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review, the author studies interactions among vertex operators, Grassmannians, and Hilbert schemes. The infinite Grassmannian is approximated by finite-dimensional cutoffs, and a family of fermionic vertex operators is introduced as the limit of geometric correspondences on the equivariant cohomology groups with respect to a one-dimensional torus actions on the cutoffs. The author proves that in the localization basis, these operators are the fermionic vertex operators on the infinite wedge representation. Moreover, the boson-fermion correspondence, locality and intertwining properties with the Virasoro algebra are the limits of relations on the cutoffs. The author further shows that these operators coincide with the vertex operators defined by A. Okounkov and the author in an earlier work on the equivariant cohomology groups of the Hilbert schemes of points on the affine plane with respect to a special torus action. Vertex operators; Grassmannians; Hilbert schemes Carlsson, E.: Vertex operators, grassmannians, and Hilbert schemes, Comm. math. Phys. 300, No. 3, 599-613 (2010) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vertex operators; vertex operator algebras and related structures, Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds Vertex operators, Grassmannians, and Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Subword complexes were defined by \textit{A. Knutson} and \textit{E. Miller} [Adv. Math. 184, No. 1, 161--176 (2004; Zbl 1069.20026)] to describe Gröbner degenerations of matrix Schubert varieties. Subword complexes of a certain type are called pipe dream complexes. The facets of such a complex are indexed by pipe dreams, or, equivalently, by monomials in the corresponding Schubert polynomial. In [Adv. Math. 306, 89--122 (2017; Zbl 1356.14039)] \textit{S. Assaf} and \textit{D. Searles} defined a basis of slide polynomials, generalizing Stanley symmetric functions, and described a combinatorial rule for expanding Schubert polynomials in this basis. We describe a decomposition of subword complexes into strata called slide complexes. The slide complexes appearing in such a way are shown to be homeomorphic to balls or spheres. For pipe dream complexes, such strata correspond to slide polynomials. flag varieties; Schubert polynomials; Grothendieck polynomials; simplicial complexes Classical problems, Schubert calculus, Reflection and Coxeter groups (group-theoretic aspects), Simplicial sets and complexes in algebraic topology, Combinatorial aspects of simplicial complexes, \(K\)-theory in geometry Slide polynomials and subword complexes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The global uniformization problem for an analytic correspondence \(f(z,w)=0\) is the problem of finding a way to pass from the implicit description \(f(z,w)=0\) to an equivalent parametric description \(z=\varphi (t)\), \(w=\psi (t)\), where \(\varphi\) and \(\psi\) are single-valued meromorphic functions in a parameter \(t\). The article is devoted to the problem of explicit construction of a global uniformization.
The authors consider the simplest case of a uniformization of an algebraic correspondence \(f(z,w)=0\). They start with an algorithm for decomposing the polynomial \(f(z,w)\) into irreducible factors. In particular, the question of irreducibility of a given polynomial is solved. The content of the corresponding section is at least of methodological interest as a new application of the Carleman ``damping function.'' The further exposition mainly follows \textit{Eh.~I.~Zverovich} [Vestn. Beloruss. Gos. Univ. Im. V. I. Lenina, Ser. I 1991, No. 1, 36--39 (1991; Zbl 0773.30043)] and \textit{O.~B.~Dolgopolova, È.~I. Zverovich} [Boundary Value Problems, Spectral Functions and Fractional Calculus (in Russian), BGU, Minsk, 76-80 (1996)] but in more detail. In the final section, various examples of an explicit construction of a global uniformization are considered. global uniformization Compact Riemann surfaces and uniformization, Algebraic functions and function fields in algebraic geometry Explicit construction of a global uniformization for an algebraic correspondence | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Two problems are analyzed. The first one refers to the decidability of the following sentence: Given a polynomial \(f\) with integer coefficients in \(v,x,y\), there is an integer \(v\) such that for any integer \(x\) there is an integer \(y\) for which \(f(v,x,y)=0\). It is proved that this problem is coNP. The second problem refers to the decidability of the following sentence: Given \(m\) polynomials \(f_1,\ldots,f_m\) with integer coefficients in \(x_1,\ldots x_n\) and \(m\geq n\), there is a rational solution to \(f_1=\cdots =f_m=0\). It is proved that this problem can be done with the complexity class \(P^{NP^{NP}}\). Diophantine problems; generalized Riemann hypothesis; Galois groups; polynomial systems; decidability; computational arithmetic geometry Rojas, J. M.: Computational arithmetical geometry I. Sentences nearly in the polynomial hierarchy, J. comput. Syst. sci. 62, 216-235 (2001) Numerical aspects of computer graphics, image analysis, and computational geometry, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.), Analysis of algorithms and problem complexity, Computer graphics; computational geometry (digital and algorithmic aspects) Computational arithmetic geometry. I: Sentences nearly in the polynomial hierarchy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal{A}=\{A_i\}_{i\in I}\) be an mp arrangement in a complex algebraic variety \(X\) with corresponding complement \(Q(\mathcal{A})=X\setminus\bigcup_{i\in I}A_{i}\) and intersection poset \(L(\mathcal{A})\). Examples of such arrangements are hyperplane arrangements and toral arrangements, i.e., collections of codimension 1 subtori, in an algebraic torus. Suppose a finite group \(\Gamma\) acts on \(X\) as a group of automorphisms and stabilizes the arrangement \(\{A_i\}_{i\in I}\) setwise. We give a formula for the graded character of \(\Gamma\) on the cohomology of \(Q(\mathcal{A})\) in terms of the graded character of \(\Gamma\) on the cohomology of certain subvarieties in \(L(\mathcal{A})\). Macmeikan, C., The Poincaré polynomial of an MP arrangement, Proc. Am. Math. Soc., 132, 1575-1580, (2004) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) The Poincaré polynomial of an mp arrangement | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems No review copy delivered. Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry A new construction of block codes from algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{J. Tate} and \textit{F. Oort} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 3, 1-21 (1970; Zbl 0195.50801)] have classified all the group schemes of rank p over rather general base rings. The next step - a description of group schemes \(G\) of period \(p\) with an action of \({\mathbb{F}}_{q}\), rk \(G=q\), was done by \textit{M. Raynaud} [Bull. Soc. Math. Fr. 102, 241-280 (1974; Zbl 0325.14020)]. This paper contains a further generalization of these results. Let \({\mathcal O}\) be a complete discrete valuation ring of characteristic 0 with residue field of characteristic \(p>0\) and containing a root of unity of order \(p\). The author studies a category \({\mathfrak G}\) of finite commutative group schemes \(G\) defined over \({\mathcal O}\) and such that rk \(G=\# G({\mathcal O})\). It is assumed also that \(G\) has period \(p\). It is introduced a category \({\mathfrak S}\) which objects are pairs \((\Gamma,\Lambda)\) where \(\Gamma\) is a finite abelian group of period \(p\) and \(\Lambda\) is an \({\mathcal O}\)-submodule in Hom\((\Gamma,{\mathcal O}/p{\mathcal O})\). The pairs \((\Gamma,\Lambda)\) must satisfy additional conditions which are related with the ''Frobenius operation'' in Hom\((\Gamma,{\mathcal O}/p{\mathcal O})\). The category \({\mathfrak S}\) is abelian and has a duality. If \(G\in {\mathfrak G}\) then the pair \((G({\mathcal O}),\Lambda(G))\), where \(\Lambda(G)\) contains all functions \(f\) from the group algebra of \(G\) such that \(f \)is a homomorphism modulo \(p{\mathcal O}\), belongs to \({\mathfrak S}\). The main result is an antiequivalence of these two categories. finite group scheme; group scheme of finite period; Frobenius operation Abrashkin, V., Group schemes of period \(p\), Math. USSR Izvestiya, 20, 411-433, (1983) Group schemes Group schemes of period \(p\). | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(GA_n\) be the affine Cremona group of the affine space \(A_n\) over a field \(K\) (or the automorphism group of the polynomial algebra \(K[x_1,\ldots,x_n]\)) and let \(TGA_n\) be the subgroup of tame automorphisms. The author calls an automorphism standard 1-parabolic if, up to a conjugation with an affine automorphism, it is of the form \((f_1,\ldots,f_{n-1},x_n+f_n)\), where the polynomials \(f_1,\ldots,f_{n-1},f_n\) do not depend on \(x_n\). The automorphism is biparabolic if it is a composition of two 1-parabolic automorphisms.
The first result of the paper under review is that \(TGA_n\) is generated by the affine group and one more, arbitrary, nonlinear standard 1-parabolic automorphism. In the special case \(n=3\) the author shows that \(TGA_3\) is generated by the affine group and an arbitrary biparabolic automorphism. automorphisms of polynomial algebras; parabolic maps; Cremona group Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Birational automorphisms, Cremona group and generalizations, Morphisms of commutative rings Generating properties of biparabolic invertible polynomial maps in three variables | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here we study the interpolation of multi-homogeneous polynomial defined over a subring of \(\mathbb{C}\). Projective techniques in algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials Ring-theoretic interpolation for polynomials | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems An order ideal \(\mathcal O\) in the polynomial ring \(K[x_1,\dots,x_n]\) in \(n\) variables over a field \(K\) is a set of terms \(\sigma=x_1^{\alpha_1}\dots x_n^{\alpha_n}\) such that any term \(\tau\) dividing \(\sigma\) belongs to \(\mathcal O\).
If \(\mathcal O\) is a finite order ideal, then the border basis scheme \(\mathbb B_{\mathcal O}\) is the affine scheme which parameterizes all the Artinian ideals \(I\) in \(K[x_1,\dots,x_n]\) with a border basis with respect to \(\mathcal O\) or, equivalently, such that \(\mathcal O\) is a basis of the quotient \(K[x_1,\dots,x_n]/I\) as a \(K\)-vector space. It is well-known that \(\mathbb B_{\mathcal O}\) can be embedded as an open subscheme in the Hilbert scheme of the \(0\)-dimensional schemes in \(\mathbb P^n_K\) with Hilbert polynomial \(p(z)=\vert \mathcal O \vert\), and that its defining ideal \(I(\mathbb B_{\mathcal O})\) is generated by quadratic polynomials.
The monomial ideal \(J\) that is generated by the terms outside \(\mathcal O\) corresponds to the origin of the affine space in which \(\mathbb B_{\mathcal O}\) is described as affine scheme. Thus, the dimension of the tangent space at \(J\) to \(\mathbb B_{\mathcal O}\), and hence to the Hilbert scheme, is computable from the generators of \(I(\mathbb B_{\mathcal O})\). More precisely, the cotangent space at \(J\) is computable. As a consequence, informations about the regularity of the border scheme and of the Hilbert scheme at \(J\) can be obtained. Thus, up to suitable changes of variables, the regularity of every point of the border basis scheme can be investigated.
The paper under review presents an efficient algorithm that computes a basis of the cotangent space at \(J\).
Relying on previous papers of the first author with A. Kehrein or L. Robbiano, this algorithm arises from an analysis of the relations among the generators of the ideal \(J\) that directly provides the linear part of the quadratic generators of \(I(\mathbb B_{\mathcal O})\) (Proposition 2.7 and Corollary 2.8). The paper ends with a brief comparison with a method introduced by \textit{M. E. Huibregtse} in an unpublished paper of 2005 [``The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space'', \url{arXiv:math/0506575}].
Other techniques and applications with analogous aims have been described in the setting of Gröbner schemes and in the term-order free setting of marked schemes over Pommaret bases, which give an open cover of Hilbert schemes with any Hilbert polynomial. For the Gröbner schemes, see [\textit{P. Lella} and \textit{M. Roggero}, Rend. Semin. Mat. Univ. Padova 126, 11--45 (2011; Zbl 1236.14006)], where a direct computation of the tangent space is provided. For the marked schemes, see section 1 of the paper [\textit{C. Bertone}, \textit{F. Cioffi} and \textit{M. Roggero}, ``Smoothable Gorenstein points via marked bases and double-generic initial ideals'', Exp. Math. (to appear), \url{doi:10.1080/10586458.2019.1592034}]. order ideal; border basis; border basis scheme; monomial point; regularity criterion; cotangent space Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Parametrization (Chow and Hilbert schemes), Computational aspects in algebraic geometry, Structure, classification theorems for modules and ideals in commutative rings On the regularity of the monomial point of a border basis scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a field of characteristic zero. It is well-known [cf. \textit{D. M. Snow} in: Topological methods in algebraic transformation groups, Proc. Conf., New Brunswick 1988, Prog. Math. 80, 165-176 (1989; Zbl 0708.14031)] that studying algebraic \(G_a\)-actions on the affine space \(\mathbb{A}^n\) (over the field \(k)\) is equivalent to studying locally nilpotent derivations \(D:B\to B\), where \(B\) is the polynomial ring in \(n\) variables over \(k\) (abbreviated \(B=k^{[n]})\). Hence, much effort has gone into attempts to understand these derivations. One way to approach this problem is to classify derivations according to their rank:
Definition. Let \(D\) be a \(k\)-derivation of \(B=k^{[n]}\). The rank of \(D\) is the least integer \(r\geq 0\) for which there exists a coordinate system \((X_1,\dots,X_n)\) of \(B\) satisfying \(k[X_{r+1}, \dots,X_n] \subseteq \ker D\). Recall that \textit{R. Rentschler} [C. R. Acad. Sci., Paris, Sér. A 267, 384-387 (1968; Zbl 0165.05402)] showed that every locally nilpotent derivation of \(k^{[2]}\) is of rank at most one. Until recently, it was not known whether, for \(n\geq 3\), locally nilpotent derivations of \(k^{[n]}\) having maximal rank \(n\) could exist; this question was answered affirmatively by \textit{G. Freudenburg} [J. Pure Appl. Algebra 126, No. 1-3, 169-181 (1998; Zbl 0904.14027)]. Note that \(\text{rank} D=n\) means that no variable of \(B=k^{[n]}\) is in \(\ker D\).
Derivations of low rank are easier to understand: \(\text{rank} D=0\) means \(D=0\), and it was shown that if \(\text{rank} D=1\) then \(D\) has the form \(f(X_2,\dots, X_n)\cdot \partial/ \partial X_1\) form some coordinate system \((X_1,\dots,X_n)\) of \(B\); in other words, \(\text{rank} D=1\) is equivalent to the two conditions \(\ker D=k^{[n-1]}\) and \(B=(\ker D)^{[1]}\). -- As to derivations of rank two, it seems that only examples and special classes have been understood. The third section of this paper is devoted to locally nilpotent derivations of \(k^{[n]}\) of rank at most two. This has the following consequence:
Corollary. Let \(D\neq 0\) be a locally nilpotent derivation of \(B=k^{[n]}\) of rank at most two. Then
(1) \(\ker D=k^{[n-1]}\).
(2) If \(D\) is fixed point free then \(D(s)=1\) for some \(s\in B\).
What we mean, here, by a fixed point of \(D\) is a fixed point of the corresponding \(G_a\)-actions on \(\mathbb{A}^n\). Hence, there do not exist fixed point free \(G_a\)-actions on \(\mathbb{A}^n\) having rank 2. However, free actions of higher rank do exist: For example, \textit{J. Winkelmann} [Math. Ann. 286, No. 1-3, 593-612 (1990; Zbl 0708.32004)] constructed a triangular \(G_a\)-action on \(\mathbb{A}^4\) which is fixed point free and of rank 3. It remains an open question whether any rank 3 algebraic \(G_a\)-action on \(\mathbb{A}^3\) can be fixed point free. -- We also point out our example 4.3 of a (rank two) non-triangulable locally nilpotent derivation of \(k[X,Y,Z]\) whose set of fixed points is a line. All results of section 3 are immediate consequences of the results of the preceding section. In section 2, we investigate locally nilpotent \(R\)-derivations of the polynomial ring \(R[X,Y]\), where \(R\) is a UFD containing the rational numbers. In particular, we give a generalization of Rentschler's theorem, a criterion for the existence of a slice and a criterion for triangulability over \(R\). fixed point of a locally nilpotent derivation; fixed point free actions Daigle, D.; Freudenburg, G., Locally nilpotent derivations over a UFD and an application to rank two locally nilpotent derivations of \(k [X_1, \ldots, X_n]\), J. Algebra, 204, 353-371, (1998) Derivations and commutative rings, Group actions on varieties or schemes (quotients), Polynomial rings and ideals; rings of integer-valued polynomials, Morphisms of commutative rings, Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial) Locally nilpotent derivations over a UFD and an application to rank two locally nilpotent derivations of \(k[X_1,\dots,X_n]\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathfrak{o}\) be a complete discrete valuation ring of mixed characteristic \((0,p)\) and \(\mathfrak{X}_0\) a smooth formal \(\mathfrak{o} \)-scheme. Let \(\mathfrak{X}\rightarrow \mathfrak{X}_0\) be an admissible blow-up. In the first part, we introduce sheaves of differential operators \(\mathcal{D}_{\mathfrak{X},k}^{\dagger}\) on \(\mathfrak{X}\), for every sufficiently large positive integer \(k\), generalizing Berthelot's arithmetic differential operators on the smooth formal scheme \(\mathfrak{X}_0 \). The coherence of these sheaves and several other basic properties are proven. In the second part, we study the projective limit sheaf \(\mathcal{D}_{\mathfrak{X},\infty}=\underset{\longleftarrow k}{\lim} \mathcal{D}_{\mathfrak{X},k}^{\dagger}\) and introduce its abelian category of coadmissible modules. The inductive limit of the sheaves \(\mathcal{D}_{\mathfrak{X},\infty}\), over all admissible blow-ups \(\mathfrak{X}\), is a sheaf \(\mathcal{D}_{\langle \mathfrak{X}_0\rangle}\) on the Zariski-Riemann space of \(\mathfrak{X}_0\), which gives rise to an abelian category of coadmissible modules. Analogues of Theorems A and B are shown to hold in each of these settings, that is, for \(\mathcal{D}_{\mathfrak{X},k}^{\dagger}, \mathcal{D}_{\mathfrak{X},\infty}\), and \(\mathcal{D}_{\langle \mathfrak{X}_0\rangle}\). Rigid analytic geometry, Rings of differential operators (associative algebraic aspects) Arithmetic structures for differential operators on formal schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies the concept of algebraic hyperstructures over the space of affine algebraic group schemes and prove that any affine algebraic group scheme \(X=\operatorname{Spec}A\) over a field \(k\) with \(|k|\geq3\) has a canonical hyperstructure induced from the coproduct on \(A\) which satisfies the weak associativity property equipped with an identity element and some other conditions. hyperfield; hyperring; hypergroup; affine algebraic group scheme; \(F_1\)-geometry Jun, Jaiung, Hyperstructures of affine algebraic group schemes, J. Number Theory, 167, 336-352, (2016) Group schemes, Hypergroups Hyperstructures of affine algebraic group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a line bundle \(L\) on the connected complex projective n-fold \(X\) there are three natural notions of the ``order'' of an embedding given by the line bundle \(L\). From strongest to weakest they are \(k\)-jet ampleness, \(k\)-very ampleness and \(k\)-spannedness. Section 1 gives a quite detailed review of the basic properties of higher order embeddings related to these notions.
In this paper the attention is concentrated mainly to the applications for \(X\) a Fano variety and \(L = -K_X\). For example, if \(X = X_{d_1,\dots,d_r} \subset {\mathbb P}^N\) is a Fano complete intersection, then theorem 2.1 shows that \(L = -K_X\) is \(k_o\)-jet ample but not \(k_o+1\) spanned, where \(k_o = (N - d_1-\dots-d_r)\). Section 3 studies the order of the embeddings of the special varieties that appear in the study of the 1-st and 2-nd order reductions of adjunction theory. By definition, the line bundle \(L\) on \(X\) is \(k\)-very ample if the restriction map \({\Gamma}(L) \rightarrow {\Gamma}({\mathcal O}_{\mathcal Z}(L))\) is surjective for any \(0\)-dimensional subscheme \({\mathcal Z}\) of \(X\) of length \(k+1\) (in particular \(0\)-very ample is equivalent to spanned on global sections and \(1\)-very ample is equivalent to very ample). In particular the \(k\)-ampleness of \(L = -K_X\) for \(k \geq 1\) yields that \(X\) is a Fano n-fold with a very ample \(-K_X\). Theorem 5.3 gives a complete description of these Fano 3-folds \(X\) for which \(L= -K_X\) is \(k\)-very ample for \(k \geq 2\). Fano threefold; Fano variety; anticanonical linear system; line bundle; ampleness; very ampleness; spannedness; higher order embeddings Mauro C. Beltrametti, Sandra Di Rocco, and Andrew J. Sommese, On higher order embeddings of Fano threefolds by the anticanonical linear system, J. Math. Sci. Univ. Tokyo 5 (1998), no. 1, 75 -- 97. Fano varieties, Embeddings in algebraic geometry, Divisors, linear systems, invertible sheaves, \(3\)-folds On higher order embeddings of Fano threefolds by the anticanoncial linear system | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In an earlier paper [\textit{R. Ragusa} and \textit{G. Zappalà}, Beitr. Algebra Geom.\ 44, No.1, 285--302 (2003; Zbl 1033.13004)], the authors introduced the notion of ``partial intersection schemes'' in projective space \(\mathbb P^r\). These schemes are \(c\)-codimensional, reduced, arithmetically Cohen-Macaulay unions of linear varieties, obtained by starting with a partially ordered subset of \(\mathbb N^c\) and carrying out a certain technical procedure. As the authors point out in the current paper (remark 1.7), their configurations are precisely the pseudo-liftings of Artinian monomial ideals, a special case of a construction by \textit{J. C. Migliore} and \textit{U. Nagel} [Commun.\ Algebra 28, No. 12, 5679--5701(2000; Zbl 1003.13005)]. Nevertheless, the authors' combinatorial approach provides a fresh and useful way of viewing these objects.
They first show that partial intersection schemes are not necessarily in the linkage class of a complete intersection (i.e.\ they are not necessarily licci). Then they give a large class of partial intersection schemes that nonetheless are licci. They complete the picture by showing that every partial intersection is in the Gorenstein linkage class of a complete intersection (i.e.\ glicci). This latter result had been shown earlier from the point of view of pseudo-liftings [\textit{J. C. Migliore} and \textit{U. Nagel}, Compos. Math.\ 133, No. 1, 25--36 (2002; Zbl 1047.14034)].
The last part of the paper gives interesting bounds and connections between the first and last graded Betti numbers of partial intersections, especially in codimension 3. A nice statement (among others) is that if \(X\) is a 3-codimensional partial intersection having \(p\) minimal last syzygies, and if \(\nu (I_X)\) is the number of minimal generators, then \(\lceil {{p+5} \over 2} \rceil \leq \nu(I_X) \leq 2p+1\). The authors also show that all possibilities in this range can occur. As they point out, such a bound is impossible in the case of arithmetically Cohen-Macaulay subschemes in general. Hilbert function; Betti numbers; liaison; arithmetically Cohen-Macaulay scheme; partial intersection schemes Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Syzygies, resolutions, complexes and commutative rings, Linkage On some properties of partial intersection schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this paper is to introduce filtrations on finite flat group schemes over unequal characteristic valuation rings which have formal properties completely parallel to those of the well-known Harder-Narasimhan filtrations on vector bundles on smooth projective curves. More generally, such filtrations are even defined and studied for families of finite flat group schemes parametrized by formal schemes.
For a smooth connected projective curve \(X\) over a field, the category \({\mathcal C}\) of vector bundles on \(X\) is exact and endowed with the two \({\mathbb Z}\)-valued functions \(\text{rank}\) and \(\text{deg}\) which are additive on short exact sequences. There is a `generic fibre' functor from \({\mathcal C}\) to the category of finite dimensional vector spaces over the function field of \(X\), and for a given vector bundle on \(X\) this functor induces a bijection between the respective subobjects. The degree function \(\text{deg}\) grows under modifications (morphisms of vector bundles which are generic isomorphisms) and detects isomorphisms (among morphisms of vector bundles which are generic isomorphisms). From these formal properties the Harder-Narasimhan filtration on vector bundles on \(X\) is defined using the function
\[
\mu=\frac{\text{deg}}{\text{rank}}.
\]
Let \(p\) be a prime number, \(K\) a complete valued field of characteristic \(0\) whose valuation \(v\) takes values in \({\mathbb R}\) and extends the \(p\)-adic valuation. Let now \({\mathcal C}\) denote the category of finite \(p\)-power order and flat commutative group schemes over \({\mathcal O}_K\) (\(=\) the valuation ring of \(K\)). One has the following two \({\mathbb R}\)-valued functions on \({\mathcal C}\): firstly, the height function \(\text{ht}\) (for \(G\in{\mathcal C}\) we have \(|G|=p^{\text{ht}(G)}\)); secondly, the degree function \(\text{deg}\) (for \(G\in{\mathcal C}\) write the conormal sheaf \(\omega_G\) as \(\omega_G=\bigoplus_i{\mathcal O}_K/a_i{\mathcal O}_K\), then \(\text{deg}(G)=\sum_i v(a_i)\): the `discriminant' of \(G\)). It is shown that \({\mathcal C}\), together with the functions \(\text{ht}\) and \(\text{deg}\) satisfies the formal properties analogous to those of the category of vector bundles on a smooth connected projective curve just recalled. Therefore, based on the function
\[
\mu=\frac{\text{deg}}{\text{ht}}
\]
one can now define a theory of Harder-Narasimhan filtrations on objects of \({\mathcal C}\) faithfully following the prototype of vector bundles on a smooth connected projective curve.
One has the corresponding Harder-Narasimhan polygons, a notion of semistability. The semistable \(G\)'s form an abelian category.
Further topics are: groups with additional structures, (kernels of) isogenies of formal \(p\)-divisible groups of dimension \(1\), families of Harder-Narasimhan filtrations over formal schemes (e.g. (semi)continuity properties of the Harder-Narasimhan polygon with respect to the topology of the generic fibre, viewed as a Berkovich analytic space), comparison with Hodge polygons, Hodge-Tate map, more general base schemes. finite flat group schemes; height; \(p\)-divisible groups; formal schemes Laurent Fargues, ``La filtration de Harder-Narasimhan des schémas en groupes finis et plats.'', J. Reine Angew. Math.645 (2010), p. 1-39 Formal groups, \(p\)-divisible groups, Group schemes The Harder-Narasimhan filtration on finite flat group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Soient \({\mathcal O} = \mathbb{C} [z_ 1, \dots, z_ n]\), \(I = (P_ 1, \dots, P_ m)\) un idéal de \({\mathcal O}\). La clôture intégrale \(\overline I\) de \(I\) est l'idéal des \(Q \in {\mathcal O}\) tels que pour tout \(z_ 0 \in \mathbb{C}^ n\) il existe \(c > 0\) et un voisinage de \(z_ 0\) sur lequel \(| Q(z) | \leq c\;\max_ i | P_ i (z) |\). Le théorème de Briançon-Skoda assure \(\overline I^ \mu \subset I\) où \(\mu = \min (m,n)\). C. Berenstein et A. Yger conjecturent que pour \(Q \in \overline I\) on peut écrire
\[
Q^ e = A_ 1P_ 1 + \cdots + A_ m P_ m
\]
où \(e = e(n)\) ne dépend que de \(n\) et \(\max_ i d^ 0A_ iP_ i \leq ed^ 0Q + d^ 0P_ 1 \dots d^ 0P_ \mu\) si \(d^ 0P_ 1 \geq \cdots \geq d^ 0P_ m\).
L'A. montre par des méthodes d'analyse complexe (formules de division) que, si la variété \(V\) des zéros de \(P_ 1, \dots,P_ m\) est discrète et \(m > n = \mu\) on peut prendre \(e(n) = n\) avec \(\max_ i d^ 0A_ iP_ i \leq n^ 2 d^ 0Q + n(4n + 1)d^ 0P_ 1 \dots d^ 0P_ n\). Si, de plus, les polynômes sont à coefficients dans \(\mathbb{Q}\) il obtient également une bonne majoration des hauteurs des \(A_ i\). En comparaison remarquons que, sans hypothèse sur \(V\), F. Amoroso a montré (par des méthodes algébriques) qu'on peut prendre \(e \leq 3^ \mu\) avec \(\max_ i d^ 0A_ i P_ i \leq 3^ \mu d^ 0Q + d^ 0P_ 1 + d^ 0P_ 1 \dots d^ 0 P_ \mu\). integral closure; Brianǫn-Skoda theorem; division formula; effectivity M. Elkadi, Une version effective du thèoréme de Briancon Skoda dans le cas algébrique discret, Acta Arith., To appear. Analytic algebras and generalizations, preparation theorems, Integral representations; canonical kernels (Szegő, Bergman, etc.), Effectivity, complexity and computational aspects of algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) An effective version of the Briançon-Skoda theorem in the discrete algebraic case | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We completely describe the Fano scheme of lines \(\mathbb{F}_1(X)\) for a projective toric surface \(X\) in terms of the geometry of the corresponding lattice polygon. toric varieties; lattice polytopes; Fano schemes N. Ilten, \textit{Fano schemes of lines on toric surfaces}, Beitr. Algebra Geom., 57 (2016), pp. 751--763, . Toric varieties, Newton polyhedra, Okounkov bodies, Parametrization (Chow and Hilbert schemes), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Fano schemes of lines on toric surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a finite subgroup of \(SL(3,\mathbb C)\). The Hilbert scheme \(\text{Hilb}^G(\mathbb C^3)\) is by definition the subscheme of \(\text{Hilb}^{\mid G\mid}(\mathbb C^3)\) parametrizing \(G\)-invariant subschemes. It has been proved that \(\text{Hilb}^G(\mathbb C^3)\) is irreducible, smooth and it is a crepant resolution of \(\mathbb C^3/G\). In this paper, the authors study this Hilbert scheme when \(G\) is a non-abelian simple subgroup of \(SL(3,\mathbb C)\). There are two such subgroups, \(G_{60}\) and \(G_{168}\), of order 60 and 168 respectively. \(G_{60}\) is isomorphic to the alternating group of degree 5 and \(G_{168}\) is isomorphic to \(PSL(2,7)\). The authors are particularly interested in giving a precise description of the fibre over the origin of \(\mathbb C^3/G\). It turns out that in the first case this fibre is a connected union of four smooth rational curves and in the second one it is a union of a smooth rational curve and a doubly blown-up projective plane, with infinitely near centres. \(G\)-invariant subschemes; crepant resolution; Hilbert scheme Gomi, Y., Nakamura, I., Shinoda, K.: Hilbert schemes of G-orbits in dimension three. Asian J. Math. 4(1), 51--70 (2000; Kodaira's issue) Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Linear algebraic groups over the reals, the complexes, the quaternions Hilbert schemes of \(G\)-orbits in dimension three | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a field of characteristic zero. Let \(\mathcal{C}\) be an affine plane \(k\)-curve, which is assumed to be integral. In this chapter, we collect various results on torsion Kähler differential form \(\omega\in\Omega_{\mathcal{O}(\mathcal{C})/k}^1\) in order to study the relation between these objects and the scheme structure of the arc scheme associated with \(\mathcal{C}\). Following this study, we provide a simple algorithm based on differential algebra to compute torsion Kahler differential form on plane curves. Arcs and motivic integration, Singularities of curves, local rings, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Arc schemes of affine algebraic plane curves and torsion Kähler differential forms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most r) of other symbols of the codeword. In this paper, we introduce a construction of LRC codes on algebraic curves, extending a recent construction of the Reed-Solomon like codes with locality. We treat the following situations: local recovery of a single erasure, local recovery of multiple erasures, and codes with several disjoint recovery sets for every coordinate (the availability problem). For each of these three problems we describe a general construction of codes on curves and construct several families of LRC codes. We also describe a construction of codes with availability that relies on automorphism groups of curves. We also consider the asymptotic problem for the parameters of the LRC codes on curves. We show that the codes obtained from asymptotically maximal curves (for instance, Garcia-Stichtenoth towers) improve upon the asymptotic versions of the Gilbert-Varshamov bound for LRC codes. Barg, A.; Tamo, I.; Vladut, S., Locally recoverable codes on algebraic curves, IEEE Trans. Inf. Theory, 63, 8, 4928-4939, (2017) Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry Locally recoverable codes on algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0655.00011.]
The author classifies quasi-smooth hypersurfaces V of degree \(h\) in a weighted projective space \({\mathbb{P}}(a,b,c,1)\) belonging to some classes described below. Let
\[
\chi (T)=T^{-h}(T^ h-T^ a)(T^ h-T^ b)(T^ h-T^ c)/(T^ a-1)(T^ b-1)(T^ c-1)=T^{m_ 1}+T^{m_ 2}+...+T^{m_ r}
\]
for some integers \(m_ 1,...,m_ r\), called the exponents. The smallest exponent \(\epsilon =a+b+c-h\) has the meaning that the canonical sheaf of V is equal to \({\mathcal O}_ V(-\epsilon)\). The author studies the following classes: (i) \(\epsilon <0\) but all other exponents are positive; (ii) \(\epsilon <0\) with some other exponents equal to 0; (iii) \(\epsilon =-2.\)- This includes the classification of such surfaces with \(\epsilon =-1.\)
As was shown earlier by the reviewer there are 31 types of such surfaces corresponding to the 31 types of Fuchsian cocompact groups whose algebra of automorphic forms is generated by three elements [Funct. Anal. Appl. 9, 149-151 (1975); translation from Funkts. Anal. Prilozh. 9, No.2, 67-68 (1975; Zbl 0321.14003) and in Group actions and vector fields, Proc. Pol.-North Am. Semin., Vancouver 1981, Lect. Notes Math. 956, 34-71 (1982; Zbl 0516.14014)]. They are all K3-surfaces with double cyclic singularities. Class (i) consists of 49 types, including 22 types corresponding to the Fuchsian groups of genus 0. The smallest exponent takes the value from the set \(\{-1,-2,-3,-4,-7\}\). There are 12 types of surfaces in class (ii), including 9 types corresponding to the Fuchsian groups of positive genus. Here \(\epsilon\in \{-1,-2,-3\}\). Finally, there are 21 types of surfaces in class (iii). There are 7 types defining K3- surfaces, 8 types corresponding to elliptic surfaces and the remaining 6 types are Horikawa's surfaces of general type. Bibliography; hypersurfaces in weighted projective space; K3-surfaces with double cyclic singularities K. Saito, Algebraic surfaces for regular systems of weights, to appear, preprint RIMS-563. Families, moduli, classification: algebraic theory, Projective techniques in algebraic geometry, Complete intersections Algebraic surfaces for regular systems of weights | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Interpolation is a central tool in numerical analysis and in computer-aided geometric design (CAGD) and scientific applications. Especially in imaging in CAGD and in physics applications for instance, not only uni- but also multivariate (sometimes also called multi-variable) interpolation is very frequently used. In this paper this is described in the context of (bivariate) surfaces (therefore most closely related to CAGD) of a method based on contraction mappings where the sought interpolants (here in particular: surfaces) are limits, so-called interpolating fractal surfaces (IFS). Convergence results and existence theorems are established especially for rational fractal interpolating surfaces. fractals; iterated function systems; fractal interpolation functions; self-referential; fractal interpolation surfaces; blending function; convergence; computer-aided geometric design Chand, AKB; Vijender, N, A new class of fractal interpolation surfaces based on functional values, Fractals, 24, 1-17, (2016) Numerical interpolation, Fractals, Computational aspects of algebraic surfaces, Interpolation in approximation theory, Multidimensional problems A new class of fractal interpolation surfaces based on functional values | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, a new heuristic symbolic-numerical method to derive exact form of the generators of the ideals in minimal prime decomposition of the radical of an ideal is presented. We set up the method without monodromy grouping. Application of the method on cyclic 9-roots polynomial system is given. A proof of the primality of the ideals is presented. Among many proved results, we also consider the residue class field of a typical prime ideal as the collection of well defined quotient of the elements in the direct sum \(\displaystyle\bigoplus_{i=7}^9 x_i\mathbb{C}[x_i] \oplus \eta \mathbb{C}[x_7,x_{\dots} \dots\delta \mathbb{C}[x_8,x_9] \oplus \sigma \mathbb{C}[x_7,x_9] \oplus \mathbb{C},\) where \(\eta=x_7x_8\), \(\delta=x_8x_9\) and \(\sigma=x_7x_9\). computational algebraic geometry; components of solutions; irreducible decomposition; symbolic-numerical algorithm; cyclic \(n\)-roots Sabeti, R.: Scheme of cyclic 9-roots. A heuristic numerical-symbolic approach. Bull. Math. Soc. Sci. Math. Roum. Tome \textbf{58}(106), 199-209 (2015) Polynomials, factorization in commutative rings, Numerical computation of solutions to systems of equations, Computational aspects of higher-dimensional varieties, Symbolic computation and algebraic computation Scheme of cyclic 9-roots. A heuristic numerical-symbolic approach | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\Lambda\) be a finite dimensional algebra over an algebraically closed field. In this paper the authors study the variety obtained from a family of objects in the derived category of \(\Lambda\)-modules with fixed dimensions or even fixed isomorphism type in homology. This affine variety extends to an algebraic scheme and the authors show that it satisfies the Grunewald-O'Halloran condition. variety of polydules; affine algebraic scheme; affine algebraic group scheme; Grunewald-O'Halloran condition Derived categories, triangulated categories, Group actions on varieties or schemes (quotients), Representations of associative Artinian rings The properties of variety of polydules as affine algebraic scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{G. Ottaviani} [Ann. Sc. Norm. Pisa, Cl. Sci., IV. Ser. 19, No. 3, 451--471 (1992; Zbl 0786.14026)] proved that, in \({\mathbb{P}}^5\), the only smooth 3-dimensional scrolls over a smooth surface are the Segre scroll, the Bordiga scroll, the Palatini scroll and the \(K3\)-scroll. The first two scrolls are arithmetically Cohen-Macaulay, and so the description of their Hilbert schemes is contained in an article of \textit{G.Ellingsrud} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 8, 423--431 (1975; Zbl 0325.14002)].
Extending methods used by \textit{G. Castelnuovo} [Ven. 1st. Atti (7) II, 855--901 (1891; JFM 23.0865.01)] in the study of Veronese surfaces, in the paper under review the authors examine the Hilbert scheme of Palatini scrolls \(X\). They prove that such a scheme has an irreducible component containing \(X\), which is birational to the Grassmannian \(G(3,\check{\mathbb{P}}^{14})\), and determine the exceptional locus of the birational map. Hilbert scheme; Grassmannian degeneracy locus; webs of linear complexes; linear system; Palatini scroll; secant variety M. L. Fania, E. Mezzetti, On the Hilbert scheme of Palatini threefolds. \textit{Adv. Geom}. 2 (2002), 371-389. MR1940444 Zbl 1054.14052 \(3\)-folds, Parametrization (Chow and Hilbert schemes), Low codimension problems in algebraic geometry, Varieties of low degree, Adjunction problems On the Hilbert scheme of Palatini threefolds. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Denoting by \(W^ r_ d(C)\) the variety of special linear systems on a given algebraic curve C of genus g, in case \(d\leq g+r-2\), \(\dim (W^ r_ d(C))\leq d-2r\) for any d and \(r\geq 1\) such that \(d\leq g+r-2\) and furthermore, equality holds if and only if C is hyperelliptic [cf. \textit{H. H. Martens}, J. Reine Angew. Math. 227, 111-120 (1967; Zbl 0172.463)]. One can then ask for a description of those non-hyperelliptic curves which achieve the maximum value d-2r-1 of \(W^ r_ d(C)\). This was given by \textit{D. Mumford} [in Contributions to Analysis, Collect, Papers dedic. L. Bers, 325-350 (1974; Zbl 0172.463)] who showed that for a non-hyperelliptic curve C, if \(\dim (W^ r_ d(C))=d-2r-1\) for some d and \(r\geq 1\) such that \(d\leq g+r-3\) then C is either a trigonal curve, a double cover of an elliptic curve, or a smooth plane curve of degree \( 5\).
The basic idea behind the theorems of Mumford and Martens is that while \(\dim (W^ r_ d(C))\leq d-2r-1\) for any non-hyperelliptic curve C, for the range of d and r such that \(\dim (W^ r_ d(C))\) is greater than the expected value \(\rho (g,r,d)=g-(r+1)(g-d+r)\). C must be a special curve in the sense of moduli [cf. \textit{P. Griffiths} and \textit{J. Harris}, Duke Math. J. 47, 233-272 (1980; Zbl 0446.14011)]. Along the lines of this idea, one of the principal objects of this paper is to establish the generalization of the theorems of Martens and Mumford and to describe explicitly those curves C for which the dimension of \(W^ r_ d(C)\) is relatively close to its maximum possible value. It is proved that: if \(\dim (W^ r_ d(C))\geq d-2r-2\leq 0\) for some d and \(r\geq 1\) such that \(d\leq g+r-4\) with \(g\geq 9\), then C is either hyperelliptic, trigonal, a 4-sheeted cover of \({\mathbb{P}}^ 1\), a double cover of an elliptic curve, a double cover of a curve of genus 2, or a smooth plane curve of degree 6. special linear system on an algebraic curve Keem, C.: On the variety of special linear systems on an algebraic curve. Math. Ann.288, 309--322 (1990) Divisors, linear systems, invertible sheaves, Curves in algebraic geometry On the variety of special linear systems on an algebraic curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a number field. It is well known that, given a positive number \(X\), the number of isomorphism classes of finite extensions \(L/K\) with fixed degree and absolute value of the norm of the discriminant \(\delta(L/K):= |N_{L/Q}D(L/K)|< X\) is finite. Let \(G\leq S_n\) be a transitive group and let \(S:= \text{Stab}_G(1)\) be the stabilizer in \(G\) of \(1\in\{1,\dots, n\}\). In [Exp. Math. 13, No. 2, 129--135 (2004; Zbl 1099.11065)] \textit{G. Malle} conjectures an asymptotic formula for the number of isomorphism classes of \(G\)-extensions \(L/K\) of a fixed number field \(K\) with \(\delta(L^S/K)< X\), where \(L^S\) is the fixed field by \(S\). Recently \textit{J. Klüners} [C. R., Math., Acad. Sci. Paris 340, No. 6, 411--414 (2005; Zbl 1083.11069)] gave a counterexample to this conjecture for \(K=\mathbb{Q}\) and \(G\) a subgroup of \(S_6\).
In the paper under review, the author considers the conjecture, with evident modifications, for a function field as a base field. Let \({\mathcal Z}(X)\) be the number of degree \(n\) extensions of \(K=\mathbb{F}_q(t)\) with some specified Galois group and with discriminant bounded by \(X\). The problem of computing the asymptotics for \({\mathcal Z}(X)\) can be related to a problem of counting \(\mathbb{F}_q\)-rational points on certain Hurwitz spaces. \textit{J. S. Ellenberg} and \textit{A. Venkatesh} [Prog. Math. 235, 151--168 (2005; Zbl 1085.11057)] used this idea to develop a heuristic for the asymptotic behavior of \({\mathcal Z}(X)\) the number of geometrically connected extensions, and showed that this agrees with the conjecture of Malle for function fields. The author extends Ellenberg-Venkatesh's argument [loc. cit.] to consider the more complicated case of covers of \(\mathbb{P}^1\), which may not be geometrically connected, and shows that the resulting heuristic suggests a natural modification to Malle's conjecture which avoids the Klüners counterexample to the original conjecture. Hurwitz schemes; branch covers; Malle's conjecture S. Türkelli, Connected components of Hurwitz schemes and Malle's conjecture, J. Number Theory 155 (2015), 163--201. Galois theory, Coverings of curves, fundamental group Connected components of Hurwitz schemes and Malle's conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review, which represents the Ph. D. thesis of the author, deals with the problem of reduction modulo \(p\) of the Hilbert-Blumenthal varieties with \(\Gamma_0 (p)\)-level structure, where \(p\) is a fixed prime natural number. One considers special kinds of Shimura varieties, namely Shimura varieties \(Sh_C (G,X)\) for which the \(p\)-primary factor \(C_p \subseteq G(\mathbb{Q}_p)\) of the subgroup \(C \subseteq G(\mathbb{A}_f)\) is of parabolic type. For these Shimura varieties the author proves results in connection with the following two problems:
(1) Describe the local structure of the model \(M_C/ \mathbb{Z}_{(p)}\) of these Shimura varieties, and
(2) Describe the reduction modulo \(p\) of this model (with special regard to the supersingular locus of \(M_C \otimes\mathbb{F}_p)\). reduction modulo \(p\); Hilbert-Blumenthal varieties; Shimura varieties Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties On the reduction of the Hilbert-Blumenthal moduli scheme with \(\Gamma_ 0(p)\)-level structure | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems An \(m\times n\) matrix M with \(m\geq n\) is rank-deficient if and only if all of its \(n \times n\) minors vanish. This occurs if and only if there is a nonzero vector \(v \in \mathbb C^n\) with \(Mv = 0\). There are \({m\choose n}\) minors and each is a polynomial of degree \(n\) in the \(mn\) entries of \(M\). In local coordinates for \(v\), the second formulation gives \(m\) bilinear equations in \(mn+n-1\) variables, and the map \((M, v)\rightarrow M\) is a bijection over an open dense set of matrices of rank \(4n-1\). The set of rank-deficient matrices has dimension \((m+1)(n-1)\), which shows that the second formulation is a complete intersection, while the first is not if \(m >n\). The principle at work here is that adding extra information may simplify the description of a degeneracy locus. Schubert varieties in the flag manifold are universal degeneracy loci [\textit{W. Fulton}, Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)]. The authors explain how to add information to a Schubert variety to simplify its description in local coordinates.
This formulates membership in a Schubert variety as a complete intersection of bilinear equations and formulates any Schubert problem as a square system of bilinear equations. This lifted formulation is both different from and typically significantly more efficient than the primal-dual square formulation in the literature.
The motivation comes from numerical algebraic geometry which uses numerical analysis to represent and manipulate algebraic varieties on a computer. It does this by solving systems of polynomial equations and following solutions along curves. For numerical stability, low degree polynomials are preferable to high degree polynomials. More essential is that Smale's \(\alpha\)-theory enables the certification of computed solutions to square systems of polynomial equations, and therefore efficient square formulations of systems of polynomial equations are desirable.
Square formulations of Schubert problems also enable the certified computation of monodromy. Since general degeneracy loci are pullbacks of Schubert varieties, these square formulations may lead to formulations of more general problems involving degeneracy loci as square systems of polynomials. Schubert calculus; certification; square systems Hein, N.; Sottile, F.: A lifted square formulation for certifiable Schubert calculus. J. symb. Comput. 79, 594-608 (2017) Classical problems, Schubert calculus, Effectivity, complexity and computational aspects of algebraic geometry A lifted square formulation for certifiable Schubert calculus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This expository paper is the content of the author's talk at the international conference on ``Zero-dimensional schemes'' in Ravello, 1992. He gives an overview of several important topics in the subject of zero-schemes. He discusses the (projectivized) tangent cone to an affine curve with an ordinary singularity at the origin, various uniformity conditions, the ideal generation and minimal resolution conjectures, the problem of finding hypersurfaces with prescribed singularities at a finite set of points, ``fat points'', and embeddings of blow-ups of \(\mathbb{P}^ 2\). He also gives several examples, describes results obtained by himself and others on these topics, and discusses open problems. ideal generation conjectures; fat points; zero-schemes; tangent cone; minimal resolution conjectures Schemes and morphisms, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities of curves, local rings Zero-dimensional schemes: Singular curves and rational surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a conjecture in Diophantine geometry concerning the construction of line bundles over smooth projective varieties over \(\overline{\mathbb{Q}}\). This conjecture, closely related to the Grothendieck period conjecture for cycles of codimension 1, is also motivated by classical algebraization results in analytic and formal geometry and in transcendence theory. Its formulation involves the consideration of \(D\)-group schemes attached to abelian schemes over algebraic curves over \(\overline {\mathbb{Q}}\). We also derive the Grothendieck period conjecture for cycles of codimension 1 in abelian varieties over \(\overline{\mathbb {Q}}\) from a classical transcendence theorem à la Schneider-Lang. algebraization; transcendence; \(D\)-group schemes; Abelian schemes Varieties over finite and local fields, Transcendence (general theory), Differential algebra, Formal neighborhoods in algebraic geometry, de Rham cohomology and algebraic geometry, Arithmetic ground fields for abelian varieties Algebraization, transcendence, and \(D\)-group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review deals with Arakelov theory on arithmetic surfaces, and especially with a generalisation of the Hodge-index theorem. Namely the main result is an upper bound for the self-intersection of a nef Cartier divisor. The proof uses the Zariski decomposition developed previously by the author [Publ. Res. Inst. Math. Sci. 48, No. 4, 799--898 (2012; Zbl 1281.14017)]. Arakelov theory; arithmetic surfaces Moriwaki, Atsushi, Numerical characterization of nef arithmetic divisors on arithmetic surfaces, Ann. Fac. Sci. Toulouse Math. (6), 0240-2963, 23, 3, 717-753, (2014) Arithmetic varieties and schemes; Arakelov theory; heights, Heights, Arithmetic ground fields for surfaces or higher-dimensional varieties Numerical characterization of nef arithmetic divisors on arithmetic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A\) be an abelian variety, \(\hat A\) its dual abelian variety and \(\mathcal P\) the normalized Poincaré line bundle on \(A\times \hat A\) so that \(\mathcal P\) parametrizes deformations of the trivial line bundle \(\mathcal O _A\). The Fourier-Mukai transform is an isomorphism between the derived categories of \(A\) and \(\hat A\) given by \(\mathbf{Rq}_*(p^*(\cdot )\otimes \mathcal P ):\mathbf D (A)\to \mathbf D (\hat A)\) where \(p:A\times \hat A \to A\) and \(q:A\times \hat A \to \hat A\) are the projections. In recent years the Fourier-Mukai functor has proven to be an invaluable tool in studying the geometry of irregular varieties, that is varieties with a non-trivial morphism to an abelian variety. In the paper under review the author revisits some of the basic results of the theory of irregular varieties and gives simplified proofs of several important theorems on the geometry of irregular varieties due to Chen-Hacon and to Ein-Lazarsfeld irregular varieties; Fourier-Mukai; generic vanishing Pareschi, Giuseppe, Basic results on irregular varieties via Fourier-Mukai methods.Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ. 59, 379-403, (2012), Cambridge Univ. Press, Cambridge Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Rational and birational maps, Vanishing theorems in algebraic geometry Basic results on irregular varieties via Fourier-Mukai methods | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(H_{d,g} (\mathbb P^r)\) denote the Hilbert scheme parametrizing curves \(C \subset \mathbb P^r\) of degree \(d\) and arithmetic genus \(g\) and let \({\mathcal I}_{d,g,r} \subset H_{d,g} (\mathbb P^r)\) be the union of irreducible components whose general member is a smooth, irreducible, non-degenerate curve. \textit{L. Ein} showed that \({\mathcal I}_{d,g,r}\) is irreducible if \(d \geq g+r\) when \(r = 3\) [Ann. Sci. Éc. Norm. Supér. 19, 469--478 (1986; Zbl 0606.14003)] and \(r=4\) [\textit{L. Ein}, Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], but there are various examples showing reducibility of \({\mathcal I}_{d,g,r}\) when \(d \geq g+r\) and \(r > 4\), disproving a claim of Severi.
Most of these examples were constructed with families of curves that are \(m\)-fold covers of \(\mathbb P^1\) with \(m \geq 3\), but the authors gave an example with a family of curves that are double covers of irrational curves [Taiwanese J. Math. 21, 583--600 (2017; Zbl 1390.14019)].
Here the authors reconstruct their example in a more geometric way as a family \(\mathcal D\) of curves on ruled surfaces. The new construction allows them to show that \(\mathcal D\) is generically smooth of expected dimension, hence a regular component. When including the distinguished component dominating \(\mathcal M_g\), this gives the first examples of Hilbert schemes \({\mathcal I}_{d,g,r}\) satisfying \(d \geq g+r\) with \textit{two} regular components. Hilbert scheme of smooth connected curves; regular components; ruled surfaces; double covers Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) Components of the Hilbert scheme of smooth projective curves using ruled surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0635.00006.]
The question addressed in the paper is: If a smooth irreducible projective curve is cut out scheme theoretically by quadrics, is then its defining ideal generated by quadratic forms? Perhaps not surprisingly so in such a general setup, the answer turns out to be negative without further restriction, but the authors provide a variety of conditions under which one does obtain the desired conclusion. In particular, this is the case for curves on two-dimensional rational normal scrolls, for curves in \({\mathbb{P}}^ 4\), and for projectively normal curves in \({\mathbb{P}}^ 5\). In higher dimensions the problem remains open for projectively normal curves, and this condition is shown to be essential. Indeed, it is proved by nice argument that the general elliptic octic in \({\mathbb{P}}^ 5\) is cut out scheme theoretically by 5 quadrics, but requires 2 more cubic generators for its defining ideal. The authors also consider versions of the problem, in which quadrics are replaced by forms of higher degree, or curves by varieties of other dimension. They show that a general set of d points in \({\mathbb{P}}^ r\), such that \((2/3)\binom{r+2}{2}< d \leq \binom{r+1}{2}\) (e.g., 15 general points in \({\mathbb{P}}^ 5)\), is scheme theoretically but not homogeneously cut out by quadrics. defining ideal generated by quadrics; projectively normal curves Ein, L.; Eisenbud, D.; Katz, S., Varieties cut by quadrics: scheme-theoretic versus homogeneous generation of ideals, (), 51-70, (Sundance, 1986) Complete intersections, Projective techniques in algebraic geometry, Curves in algebraic geometry Varieties cut out by quadrics: Scheme-theoretic versus homogeneous generation of ideals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A modified hypergeometric system is a system of linear partial differential equations. It was introduced for studying solutions of hypergeometric systems along a curve \((y_1, \dots, y_n) = (c_1 t^{w_1}, \dots, c_n t^{w_n})\). A hypergeometric system \(H_A(\beta)\) is determined by an integer \(d \times n\) matrix \(A\) and a complex vector \(\beta \in \mathbb C^d\). In addition, a modified hypergeometric system \(H_{A, w, \alpha}(\beta)\) is determined by an integer vector \(w = (w_1, \dots, w_n)\) and a complex number \(\alpha\).
Let \(\mathcal D\) be the sheaf of holomorphic differential operators on \(X = \mathbb C^{d+1}\),
\(\mathcal M_{A, w, \alpha}(\beta) = \mathcal D/\mathcal D H_{A, w, \alpha}(\beta)\), and \(\beta \in \mathbb C^d\) very generic. The authors show that \(\mathcal M_{A, w, \alpha}(\beta)\) does not have non-zero solutions for very generic \(\alpha \in \mathbb C\), and compute the dimension and a basis of the solution space of \(\mathcal M_{A, w, \alpha}(\beta)\) if it has non-zero solutions. On the other hand, they also compute the Gevrey solutions of \(\mathcal M_{A, w, \alpha}(\beta)\) modulo convergent power series, when \(\alpha\) is very generic.
Later they study a Laplace integral representation of a solution of \(\mathcal M_{A, w, \alpha}(\beta)\) by the Borel summation method. As an application, they give an analytic meaning to the Gevrey solution of the hypergeometric system \(H_A(\beta)\). hypergeometric systems; Gevrey series solutions; Laplace integral representation Sheaves of differential operators and their modules, \(D\)-modules, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Other hypergeometric functions and integrals in several variables, Commutative rings of differential operators and their modules, Toric varieties, Newton polyhedra, Okounkov bodies Irregular modified \(A\)-hypergeometric systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We propose some cryptographic algorithms based on a finite \(BN\)-pair \(G\) defined over the field \(F_q\). We convert the adjacency graph for maximal flags of the geometry of the group \(G\) into a finite Tits automaton by special colouring of arrows and treat the largest Schubert cell \(\text{Sch}=(F_q)^N\) on this variety as a totality of possible initial states and a totality of accepting states at a time. The computation (encryption map) corresponds to some walk in the graph with the starting and ending points in Sch. To make algorithms fast, we will use the embedding of the geometry for \(G\) into the Borel subalgebra of the corresponding Lie algebra. We consider the induced subgraph of the adjacency graph obtained by deleting all vertices outside the largest Schubert cell and the corresponding automaton (Schubert automaton). We consider the following symbolic implementation of Tits and Schubert automata. The symbolic initial state is a string of variables \(x_\alpha\), where the roots \(\alpha\) are listed according to the Bruhat order, choice of the label will be governed by a linear expression in the variables \(x_\alpha\), where \(\alpha\) is a simple root. Conjugations of such a nonlinear map with elements of an affine group acting on \((F_q)^N\) can be used in the Diffie-Hellman key-exchange algorithm based on the complexity of the group-theoretical discrete logarithm problem in case of the Cremona group of this variety. We evaluate the degree of these polynomial maps from above and the maximal order of this transformation from below. For simplicity we assume that \(G\) is a simple Lie group of normal type, but the algorithm can be easily generalised to wide classes of Tits geometries. In the spirit of algebraic geometry we slightly generalise the algorithm by change of the linear governing functions for rational linear maps. small world graphs; Lie geometries; symbolic computation; walks on graphs; Schubert cells; cryptography; key-exchange protocols; Tits automaton; Schubert automaton [2] Ustimenko V., ''Schubert cells in Lie geometries and key exchange via symbolic computations'', Proceedings of the International Conference ''Applications of Computer Algebra'', ACA 2010 (Vlora, 2010), Albanian Math. J., 4, no. 4, 2010, 135--145 Cryptography, Small world graphs, complex networks (graph-theoretic aspects), Applications of graph theory, Birational automorphisms, Cremona group and generalizations, Groups with a \(BN\)-pair; buildings Schubert cells in Lie geometries and key exchange via symbolic computations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\tau\) be a topological type for nodal projective curves and \(M[\tau]\) the variety of all nodal curves with \(\tau\) as topological type. Here we prove that for many \(\tau\) and many multidegrees a general \(X\in M[\tau]\) has a a generically smooth component with the expected dimension of the Brill-Noether scheme of morphisms \(X\to\mathbb P^n\), \(n\geq 3\), with the prescribed multidegree. stable curve; Brill-Noether theory; restricted tangent bundle; multidegree Families, moduli of curves (algebraic), Plane and space curves, Special divisors on curves (gonality, Brill-Noether theory) Good components of the Brill-Noether scheme for general stable curves with fixed topological type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(V = {\mathbb C}^n\) be an \(n\)-dimensional complex vector space. For a sequence of integers \(\{i_1, i_2, \dots, i_m\}\) with \(0 < i_1 < i_2 < \cdots < i_m = n\), a partial flag of \(V\) of type \((i_1,i_2,\dots, i_m)\) is a sequence of subspaces \(F_{i_1} \subset F_{i_2} \subset \cdots \subset F_{i_m}\) with \(\text{dim}F_{i_j} = i_j\). A complete (or full) flag is one of type \((1,2,\dots, n)\). Consider a nilpotent endomorphism \(N : V \to V\). The variety of all complete flags that are invariant under \(N\) is known as the Springer fiber of \(N\). More generally, the variety of \(N\)-invariant partial flags of some type \((i_1,i_2,\dots,i_m)\) is known as a Spaltenstein variety of type \((i_1,i_2,\dots,i_m)\). The nilpotent endomorphism \(N\) is determined by the sizes of its Jordan blocks and hence can be associated with a partition \(\lambda\) of \(n\). Spaltenstein showed that the irreducible components of the Springer fiber are in one-to-one correspondence with standard Young tableaux of shape \(\lambda\). More generally, the irreducible components of a Spaltenstein variety (of a given type) can also be (bijectively) associated with a set of tableaux determined by the type.
The goal of this paper is to study the geometry of the irreducible (and so-called ``generalized irreducible'') components of a Spaltenstein variety in the case when \(N\) has two Jordan blocks by generalizing known results for the special case of the Springer fiber. The author first gives an explicit description of these irreducible components by generalizing the description given for Springer fibers by \textit{F. Fung} [Adv. Math. 178, No. 2, 244--276 (2003; Zbl 1035.20004)]. Following \textit{C. Stroppel} and \textit{B. Webster} [Comment. Math. Helv. 87, No. 2, 477--520 (2012; Zbl 1241.14009)], associated to a row-strict tableaux of a given type, the author defines the notion of a ``generalized irreducible component'' of a Spaltenstein variety (of that given type). The collection of generalized irreducible components contains the honest irreducible components. Two key tools are used in the paper: the ``dependence graph'' associated to a tableaux and the ``circle diagram'' associated to a pair of tableaux. Dependence graphs extend the notion of cup diagrams used in the work of Fung and Stroppel-Webster while circle diagrams (which are built out of two extended cup diagrams) were introduced for Springer fibers by \textit{C. Stroppel} [Compos. Math. 145, No. 4, 954--992 (2009; Zbl 1187.17004)].
The first main result is a graphical description of generalized irreducible components in terms of the associated dependence graph. This is also extended to the intersection of a pair of such components. Circle diagrams can be used to determine when the intersection of two generalized irreducible components is empty. By showing that generalized irreducible components and (non-empty) intersections of pairs of such can be identified with iterated fiber bundles, the author computes the cohomology of such spaces. Finally, it is shown that the direct sum over all pairs of the cohomology groups of pair-wise intersections can be given an algebra structure via colored cobordisms. Springer fiber; Spaltenstein varieties; Young tableaux; dependence graphs; cup diagrams; circle diagrams; iterated fiber bundle; cohomology; colored cobordisms DOI: 10.1017/S0017089512000110 Grassmannians, Schubert varieties, flag manifolds, Classical real and complex (co)homology in algebraic geometry, Rings arising from noncommutative algebraic geometry, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Monoidal categories (= multiplicative categories) [See also 19D23] A graphical calculus for 2-block Spaltenstein varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The integers \(n\geq m-1\geq 0\), a Birkhoff interpolation problem \({\mathcal B}\) (interpolation of a polynomial and certain of its derivatives of order \(\leq n\) at \(m\) points of a field) induced by a matrix \(E= [e_{i,k}]\), \(1\leq i\leq m\), \(0\leq k\leq n\), \(e_{i,k}\in\{0,1\}\), a prime \(p>n\) and a \(p\)-power \(q\). Here we prove the regularity of \({\mathcal B}\) at \((t_1,\dots,t_m)\in\mathbb F_q^m\) if it is regular at \((t_1^{q/p},\dots, t_1^{q/p})\in\mathbb F_p^n\). The regularity over \(\mathbb F_p\) was recently studied by T. Tassa to solve a cryptographic model (hierarchical threshold secret sharing) . Finite ground fields in algebraic geometry, Approximation by polynomials, Applications to coding theory and cryptography of arithmetic geometry, Multidimensional problems, Projective techniques in algebraic geometry Birkhoff interpolation over a finite field | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If \(X\) is a (quasi-)projective variety, there are several triangulated categories naturally associated to it: Probably the most common ones are the bounded derived category of coherent sheaves \({\text D}^{\text b}(X)\) and its full triangulated subcategory of perfect complexes \({\text{Perf}}(X)\), which is roughly the smallest triangulated subcategory of \({\text D}^{\text b}(X)\) containing all finite-rank locally free sheaves. These categories coincide if and only if \(X\) is smooth.
In recent years it has become commonplace to investigate the geometry of a smooth projective variety through its bounded derived category of coherent sheaves, so most of the results are formulated, and the proofs usually only work, under the smoothness assumption. In the article under review the author extends two such results to the singular case.
Firstly, \textit{A.\ Bondal} and \textit{M.\ van den Bergh} proved [Mosc.\ Math.\ J.\ 3, No.\ 1, 1--36 (2003; Zbl 1135.18302)] that for a smooth and proper variety \(X\) any covariant or contravariant locally-finite (a certain boundedness condition) cohomological functor from \({\text D}^{\text b}(X)\) to the category of vector spaces is representable. Furthermore, \({\text D}^{\text b}(X)\) is equivalent to either of these categories of functors. The first main result of this paper states that dropping the smoothness one has that \({\text D}^{\text b}(X)\) is equivalent to the category of locally-finite, cohomological functors on \({\text{Perf}}(X)\). The proof uses the machinery of compactly-generated triangulated categories.
Secondly, a result of \textit{A.\ Bondal} and \textit{D.\ Orlov} [Compos.\ Math.\ 125, No.\ 3, 327--344 (2001; Zbl 0994.18007)] says that if a smooth and projective variety \(X\) has ample or anti-ample canonical bundle, then any variety \(Y\) satisfying \({\text D}^{\text b}(X)\cong{\text D}^{\text b}(Y)\) is isomorphic to \(X\). The author extends this result to the case where \(X\) is projective Gorenstein using, in particular, a relativization of the notion of a Serre functor.
Besides the above mentioned results the author also proves that two projective schemes have equivalent bounded derived categories if and only if the respective categories of perfect complexes are equivalent. Furthermore, for a projective scheme \(X\) the groups of autoequivalences of \({\text D}^{\text b}(X)\) and of \({\text{Perf}}(X)\) coincide. This result is proved using so-called pseudo-adjoint functors. derived categories; compactly-generated triangulated categories; perfect complexes; saturated triangulated categories; pseudo-adjoint functors Ballard, M.: Derived categories of singular schemes with an application to reconstruction. Adv. Math. 227(2), 895--919 (2011) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories Derived categories of sheaves on singular schemes with an application to reconstruction | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems At the beginning of the 21st century A. Givental has found a recursive procedure reconstructing the full potential of the cohomological field theory from its genus zero data. Several years later a very general recursive reconstruction theory was proposed originating from the matrix models. It is now known as the Chekhov-Eynard-Orantin recursion or topological recursion. Various series of numbers that were known to have ``classical'' recursive relations (like Hurwitz numbers and Hodge integrals) were put in the framework of Chekhov-Eynard-Orantin theory too. However it was not clear at all if the reconstruction of Givental agrees with the reconstruction of the topological recursion.
In this article the authors prove that the reconstruction of Givental is an example of the Chekhov-Eynard-Orantin recursion. Using Givental's theory the authors prove the conjecture of Norbury and Scott about the topological recursion for the Gromov-Witten theory of \(\mathbb CP^1\). topological recursion; Givental action; Gromov-Witten theory P. Dunnin-Barkowski, N. Orantin, S. Shadrin, and L. Spitz, Identification of the Givental formula with the spectral curve topological recursion procedure , preprint, [math.ph]. 1211.4021v1 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Identification of the Givental formula with the spectral curve topological recursion procedure | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be an algebraically closed field of positive characteristic \(p\). A \(k\)-derivation \(d\) on \(k[x,y]\) is called of multiplicative type if \(d^ p=d\). Let \(R\) be a discrete valuation ring with residue field \(k\). \(R\) is \(p\)-good iff its quotient field is of characteristic zero and contains a primitive \(p\)-th root of unity. Let \(\mu_ p\) be the finite multiplicative group scheme over \(R\) whose coordinate ring is \(R[t,t^{- 1}]/\langle t^ p-1\rangle\). An algebraic action of \(\mu_ p\) on \(\hbox{Spec}(R[x,y])\) induces a \(k\)-derivation \(d\) on \(k[x,y]\) of multiplicative type. These derivations are called liftable.
The authors show that linearizable derivations of multiplicative type are liftable. They conjecture that linearizability is also a necessary condition for liftability. --- In the first chapter the authors classify all linearizable \(k\)-derivations of multiplicative and additive (i.e. \(d^ p=0)\) type on \(k[x_ 1,\ldots,x_ n]\). group actions on the affine plane; liftable derivations; discrete valuation ring; group scheme; linearizable derivations Group actions on varieties or schemes (quotients), Polynomial rings and ideals; rings of integer-valued polynomials, Morphisms of commutative rings, Schemes and morphisms Finite group scheme actions on the affine plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper describes a grade-theoretic analogue of the Cousin complex introduced into algebraic geometry and commutative algebra by Grothendieck [see Chapter IV of the book of \textit{R. Hartshorne}, ''Residues and duality'', Lect. Notes Math. 20 (1966; Zbl 0212.261)]; an elementary construction of certain Cousin complexes was given by the reviewer in Math. Z. 112, 340-356 (1969; Zbl 0182.061).
The author constructs, for a commutative Noetherian ring R, a grade- theoretic analogue
\[
0\to^{d^{-2}}R\to^{d^{-1}}D_ 0\to^{d^ 0}D_ 1\to...\to D_ n\to^{d^ n}D_{n+1}\to...
\]
of the Cousin complex which is such that, for \(n\geq 0\) for which there exist proper ideals of R of grade \(n+1\), \(D_ n\cong \to^{\lim}_{\text{grade}(L)\geq n+1}Hom_ R(L,\text{coker} (d^{n- 2}))\). The author hints at some applications, but commentsthat ''a mature exposition of these topics must await another occasion''. depth; Cohen-Macaulay ring; Gorenstein ring; local cohomology modules; torsion modules; grade-theoretic analogue of the Cousin complex; Noetherian ring Hughes, Quaestiones Math. 9 pp 293-- (1986) Complexes, Commutative Noetherian rings and modules, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Torsion modules and ideals in commutative rings, Local cohomology and algebraic geometry A grade-theoretic analogue of the Cousin complex | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic varieties. As a first application, we provide a general incidence estimate that is tight in its dependence on the size, degree and dimension of the varieties involved. Semialgebraic sets and related spaces, Erdős problems and related topics of discrete geometry, Real algebraic sets The polynomial method over varieties | 0 |
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