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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 We investigate a Curtis-Tits style group presentation based on the Dynkin diagram \(C_g\), in which the short root generators have order two while the long root generators have order four. We prove that this describes a finite group with an almost extraspecial normal subgroup of order \(2^{2 g + 2}\) and quotient isomorphic to the symplectic group \(\mathsf{Sp}(2 g, 2)\). Such an extension has to be non-split if \(g \geqslant 3\), and was proved to exist in papers of \textit{B. Bolt} et al. [J. Aust. Math. Soc. 2, 60--79, 80--96 (1961; Zbl 0097.01702)] and \textit{R. L. Griess jun.} [Pac. J. Math. 48, 403--422 (1973; Zbl 0283.20028)]. Our presentation proves that it is a double cover of a finite quotient of \(\mathsf{Sp}(2 g, \mathbb{Z})\). We investigate a \(2^g\) dimensional complex representation on a suitable space of theta functions, and produce some consequences for the signatures of 4-manifolds described as surface bundles over surfaces. In particular, we prove that if the monodromy is contained in the theta subgroup \(\mathsf{Sp}^{\mathsf{q}}(2 g, \mathbb{Z}) \leqslant \mathsf{Sp}(2 g, \mathbb{Z})\) then the signature of the 4-manifold is divisible by eight. finite groups; Curtis-Tits presentations; symplectic groups; theta functions Modular representations and characters, Theta functions and abelian varieties, Generators, relations, and presentations of groups, Theta series; Weil representation; theta correspondences Theta functions and a presentation of \(2^{1 +(2 g + 1)} \mathsf{Sp}(2 g, 2)\) | 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 \(Y\to X\) be a Galois covering of complex analytic spaces with Galois group \(G\) and \(X\) compact. Here the author constructs a theory of \(L^p\)-cohomology for coherent sheaves on \(Y\) equipped with a \(G\)-action. He can even handle the case in which the discrete group \(G\) acts on \(Y\) with fixed points (discrete co-compact action of \(G\) on \(Y)\) and the very important case in which on the sheaves acts a central extension of \(G\) by \(S^1\), obtaining a theory which contains the ``Vafa-Witten twisting trick'' introduced by Gromov. He obtains an Atiyah \(L^2\)-index theorem in this set-up. coherent analytic sheaves; complex spaces; \(L^2\)-cohomology groups; Galois covering; Atiyah's \(L^2\) index theorem; \(L^p\)-cohomology theory for coherent analytic sheaves; central extension by \(S^1\); discrete co-compact action Eyssidieux, Philippe, Invariants de von Neumann des faisceaux analytiques cohérents, Math. Ann., 317, 527-566, (2000) Analytic sheaves and cohomology groups, Compact complex surfaces, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Coverings in algebraic geometry, General theory of von Neumann algebras, Index theory and related fixed-point theorems on manifolds Von Neumann invariants of coherent analytic sheaves | 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 algebraic plane curve defined over the field of complex numbers, let \(P_ 1,...,P_ 2\) be the singular points of K and let \(U_ 1,..,U_ s\) be neighbourhoods of \(P_ 1,...,P_ s\), respectively. Denote by G the fundamental group of the complementary set \({\mathbb{P}}^ 2({\mathbb{C}})\setminus K\) and by \(G_ i\) the image in G of the fundamental group of \(U_ i\setminus P_ i\) via the natural inclusion map, \(i=1,...,s\). - Generalizing a result of his previous paper [Math. USSR, Sb. 65, No.1, 267-277 (1990); translation from Mat. Sb., Nov. Ser. 137(179), No.2(10), 260-270 (1988; Zbl 0662.14004)], the author proves the following theorem:
The commutant [G,G] is the minimal normal subgroup of G containing all the commutants \([G_ i,G_ i]\), \(i=1,...s\). Zariski problem; fundamental group of the complement of a plane algebraic curve S. Yu. Orevkov, The commutant of the fundamental group of the complement of a plane algebraic curve, Uspekhi Mat. Nauk 45 (1990), no. 1(271), 183 -- 184 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 1, 221 -- 222. Coverings in algebraic geometry, Special algebraic curves and curves of low genus, Derived series, central series, and generalizations for groups The commutant of the fundamental group of the complement of a plane 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 [For the entire collection see Zbl 0655.00010.]
The author investigates \(\ell\)-adic cohomology groups of Kummer coverings \(X_ n\) of the two-dimensional projective space \({\mathbb{P}}^ 2\) ramified along the complete quadrilateral. He considers the action of the Galois group \(A=Gal(X_ n/{\mathbb{P}}^ 2)\) on the cohomology space and decomposes it into eigenspaces \(H^ i_{\ell}(\alpha)\) with respect to characters \(\alpha: A\to {\bar {\mathbb{Q}}}^*_{\ell}\). The L-function associated to the abelian representation \(\rho^ 2_{\alpha}: Gal(k_ s/k)\to Aut(H^ 2_{\ell}(\alpha))\) is computed for generic symmetric \(\alpha\). \(\ell \)-adic cohomology groups of Kummer coverings; L-function Coverings in algebraic geometry, Galois cohomology, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Birational automorphisms, Cremona group and generalizations Abelian \(\ell\)-adic representations associated with Selberg integrals | 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 \(Y_ n\), \(n\geq n_ 0\), be a sequence of n-dimensional algebraic varieties over the algebraically closed field k, which are weighted complete intersections in the weighted projective space \({\mathbb{P}}(e_ 0,...,e_{n+r})\). Assume that \(Y_ n\) is the intersection of \(Y_{n+1}\) with \({\mathbb{P}}(e_ 0,...,e_{n+r})\) where \({\mathbb{P}}(e_ 0,...,e_{n+r})\) is considered as a subspace of \({\mathbb{P}}(e_ 0,...,e_{n+r+1})\) in the obvious way. Let \(X_ n\to Y_ n\) be a sequence of locally complete intersection morphisms such that \(X_ n=X_{n+1}\times_{Y_{n+1}}Y_ n\) for \(every\quad n.\) Then, generalizing a result of \textit{E. Sato}, the following is shown in the paper: If \({\mathcal O}_{Y_ n}(1)\) is invertible and \(X_ n\) is reduced and connected then \(X_ n\) is a weighted complete intersection in some projective space \({\mathbb{P}}(e_ 0,...,e_{n+r},b_ 1,...,b_ s)\) for every n. In particular, this applies to towers of coverings \(X_ n\) of the usual projective space \(Y_ n={\mathbb{P}}^ n\). For the proof the methods of a previous paper by the author [cf. Math. Ann. 271, 153-160 (1985; Zbl 0541.14011)], play an important role.
As a byproduct there is also shown a generalization of the theorem of A. N. Tyurin and E. Sato on infinitely extendable vector bundles. weighted complete intersections; weighted projective space; towers of coverings; infinitely extendable vector bundles -, Babylonian tower theorems for coverings (to appear). Complete intersections, Coverings in algebraic geometry, Projective techniques in algebraic geometry, Low codimension problems in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Babylonian tower theorems for coverings | 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 0509.00008.]
Let \(f:\quad X^ n\to {\mathbb{P}}^ p\) be a finite morphism into a projective space with \(n\leq p\). The main purpose is to investigate relations between the fundamental group of X and the set of points of some fixed multiplicity. Using a new notion of multiplicity called stable multiplicity, \(m_ S(f)(x)\), the following theorem is proved. Theorem: Let \(f:\quad X^ n\to {\mathbb{P}}^ p,\) \(n\leq p\), be a finite morphism of a projective variety analytically irreducible at every point into a projective space. Suppose that there exists a\ d such that \(d(p-n+1)\leq n\) and every component of the set \(T_ S^{d+1}(f)=\{x\in X:m_ S(f)(x)\geq d+1\}\) has codimension \(>d(p-n+1)\) or is empty. Then X is simply connected.
The notion of stable multiplicity is given as follows. Let \(f:\quad (X^ n,x)\to ({\mathbb{C}}^ p,0)\) be a proper finite-to-one complex analytic map- germ. The topological multiplicity \(m_ T(f)(x)\) of f at x is defined to be \(m_ T(f)(x)=\lim_{\epsilon \to 0}\max \{\#(f^{-1}(y)\cap B_{\epsilon}(x)):y\in B_{e}(f(x))\}.\) Let \(F:\quad (X\times {\mathbb{C}}^ r,(x,0))\to ({\mathbb{C}}^ p\times {\mathbb{C}}^ r,0\times 0)\) be a stable unfolding of f in J. Mather's sense, which always exists for a finite type map-germ. Then the stable multiplicity \(m_ S(f)(x)\) of f at x is defined to be \(m_ T(F)(x,0)\). Although when \(n=p\) the three definitions of multiplicities \(m_ T(f)(x)\), \(m_ S(f)(x)\) and dim Q(f) coincide, they differ when \(n<p\). It is pointed out that stable multiplicity has better properties than the other two in the following sense: Stable multiplicity has the additive property, although topological multiplicity does not: When \(n<p\), dim Q(f)(x) seems to count x too many times: Also \(m_ S(f)(x)\) seems to agree with the algebraic geometer's intuition as to what the multiplicity should be. - After a theorem about the codimension of \(T_ S^{d+1}(f)\) and an existence theorem for \(T_ S^{d+1}(f)\) are proved, the main theorem is proved. For related results, see papers by \textit{J. P. Hansen} [''A connectedness theorem for flagmanifolds and Grassmannians and singularities of morphisms to \({\mathbb{P}}^ m\)'', Ph. D. Thesis, Brown Univ. (Providence, R. I. 1980)], \textit{R. Lazarsfeld} [''Branched coverings of projective space'', Ph. D. Thesis, Brown Univ., [Providence, R. I. 1980)] and the author and \textit{R. Lazarsfeld} [Invent. Math. 59, 53-58 (1980; Zbl 0422.14010)]. fundamental group; stable multiplicity Gaffney, T.: Multiple points and associated ramification loci. In: Proceedings of Symposia in Pure Mathematics 40:I, Peter Orlik (ed.), American Mathematical Society, Providence 1983, pp. 429--437 Singularities in algebraic geometry, Coverings in algebraic geometry, Ramification problems in algebraic geometry, Germs of analytic sets, local parametrization Multiple points and associated ramification loci | 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 past decade, various attempts have been made to tackle the Shafarevich conjecture which predicts the holomorphically convexity of the universal covering \(\widetilde X\) of any projective manifold \(X\). From these investigations, there were some candidates for a possible counterexample for that conjecture. Then came Kollár's approach to investigate the case when the fundamental group of \(X\) admits large finite-dimensional representations. In this direction, the author shows that, for a given \(n\in\mathbb{N}\), the topological covering space \(\widetilde X_n\) of a projective algebraic manifold \(X\) corresponding to the intersection of the kernels of all linear reductive representations of \(\pi_1(X)\) to \(\text{GL}_n(\mathbb{C})\) is holomorphically convex.
When \(\dim X= 2\), this result is a corollary of a theorem due to \textit{L. Katzarkov} and \textit{M. Ramachandran} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 31, No. 4, 525--535 (1998; Zbl 0936.14012)]. Eyssidieux, P., Sur la convexité holomorphe des revêtements linéaires réductifs d'une variété projective algébrique complexe, Invent. Math., 156, 503-564, (2004) Holomorphically convex complex spaces, reduction theory, Coverings in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects), Uniformization of complex manifolds On the holomorphic convexity of reductive linear coverings of a projective complex algebraic manifold | 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 Both notions of \(k\)-jet ample line bundle and \(k\)-very ample line bundle on a smooth projective variety \(X\) have been introduced to capture the concept of higher order embedding of \(X\). The former notion requires the simultaneous separation of jets at finitely many points of \(X\), while the latter requires the surjectivity of the restriction of sections to 0-dimensional subschemes of a certain length. In the paper under review, the authors investigate higher-order embeddings of cyclic coverings \(\pi:Y\to X\) via line bundles obtained by pulling back sufficiently positive line bundles on \(X\) to \(Y\). In particular, if \(\pi\) is the cyclic covering of degree \(d\) defined by a line bundle \(M\in\text{Pic}(X)\), \(L\) is a line bundle on \(X\), and \(k\) is a nonnegative integer, they show that \(\pi^*L\) is \(k\)-jet ample provided that \(L-qM\) is \((k-q)\)-jet ample for \(q=0,\dots, \min\{k,d -1\}\). Similarly, \(\pi^*L\) is \(k\)-very ample provided that \(L-qM\) is \(\sigma\)-very ample for \(q=0, \dots, \min \{k,d-1\}\), where \(\sigma=\sigma (k,d,q)\) is a suitably defined integer, whose values are tabulated in some instances to give a clear feeling about the result. Finally, several examples are produced showing that the assumptions above cannot be dropped, and pointing out connections with the study of linear systems of plane curves with fat base points. fat base point; ample line bundle; higher-order embeddings; cyclic coverings; linear systems of plane curves Embeddings in algebraic geometry, Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry Cyclic coverings and higher order embeddings of algebraic 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 This is a very interesting paper. Given a triangulated category \(\mathcal K\) furnished with a symmetric monoidal product \(\otimes\) on \(\mathcal K\) which is exact in both variables. Then the author defines in a functorial way a `geometry' on \(\mathcal K\) by defining what is a prime ideal \(\mathfrak p\). A thick tensor ideal is a thick triangulated subcategory \(\mathcal A\) of \(\mathcal K\) so that for any \(a\) in \(\mathcal A\) and \(b\) in \(\mathcal K\) one has that \(a\otimes b\) is in \(\mathcal A\). A thick tensor ideal \(\ mathfrak p\) is prime if whenever \(a\otimes b\) is in \(\mathfrak p\), then \(a\) is in \(\mathfrak p\) or \(b\) is in \(\mathfrak p\). The prime spectrum of \(\mathcal K\) is the set of all primes in \(\mathcal K\). The support of an \(a\) in \(\mathcal K\) is the set of primes not containing \(a\).
First the author establishes elementary properties such as that this defines indeed a Zariski-like topology on \(\mathcal K\), that the spectrum is never empty, that maximal proper thick \(\otimes\)-ideals are prime, that localisation is possible, that complements of supports are quasi-compact and these are the only quasi-compacts, and similar statements.
Then, in a second section the author shows that the support of the identity object (with respect to the tensor structure) has as support the whole spectrum, the zero object has empty support, that the support of a direct sum is the union of the summands, that the support does not change by the triangulated shift-functor \(T\), that the support of an \(a\) is included in the union of the support of \(b\) and \(c\) whenever one has a distinguished triangle \(a\rightarrow b\rightarrow c\rightarrow Ta\), and that the support of \(a\otimes b\) is the intersection of the supports of \(a\) and of \(b\). The author shows that whenever one has a `support theory' on \(\mathcal K\) satisfying these properties, meaning a space \(X\) and a mapping associating to any \(a\) in \(\mathcal K\) a closed set in \(X\) satisfying these above properties, then there is a unique continuous map from \(X\) to the spectrum of \(\mathcal K\) so that taking preimages maps the corresponding closed sets to each other.
In a third section the author classifies thick subcategories of \(\mathcal K\) in a certain sense and in the final two sections the author gives two very striking applications. Using a result of Thomason, the author shows that for a topologically noetherian scheme \(X\) the spectrum of the derived category of perfect complexes over \(X\) is isomorphic to \(X\). Let \(G\) be a finite group (scheme), and \(k\) be a field of finite characteristic. The author shows that the spectrum of the stable module category of finitely generated \(kG\)-modules is isomorphic to the projective space associated to the the cohomology \(H^*(G,k)\) of the group. This last statement was independently discovered by Friedlander and Pevtsova, using completely different methods. support varieties; triangulated categories; tensor categories Paul Balmer, \emph{The spectrum of prime ideals in tensor triangulated categories}, J. Reine Angew. Math. \textbf{588} (2005), 149--168. \MR{MR2196732 (2007b:18012)} Derived categories, triangulated categories, Noncommutative algebraic geometry, Group schemes, Homological functors on modules (Tor, Ext, etc.) in associative algebras, Stable homotopy theory, spectra, Modular representations and characters, Cohomology of groups The spectrum of prime ideals in tensor triangulated categories | 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 deals with complex algebraic surfaces of general type whose Chern numbers satisfy the extremal relation \(c^ 2_ 1=3 c_ 2\) (and therefore are ball quotient surfaces). Examples have been constructed by the author before as special coverings of the projective plane ramified along special configuration of lines [see the author in Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 113-140 (1983; Zbl 0527.14033)]. If one allows different branching orders along the lines, one has a chance to find more examples. In the paper a formula for the ''deviation'' 3 \(c_ 2-c^ 2_ 1\) of such coverings is given, and restrictions on the combinatorics of suitable line configurations are deduced from the relative proportionality principle for curves on ball quotient surfaces. Finally all known examples are listed. The actual construction of coverings and the case of non-compact ball quotients (also contained in the reviewer's Ph. D. thesis) are not touched here. surfaces of general type; Chern numbers; ball quotient surfaces; coverings of the projective plane Hirzebruch, F., \textit{algebraic surfaces with extreme Chern numbers}, Russian Math. Surveys, 40, 135-145, (1985) Families, moduli, classification: algebraic theory, Special surfaces, Characteristic classes and numbers in differential topology, Coverings in algebraic geometry Algebraic surfaces with extreme Chern numbers (Report on the thesis of Th. Höfer, Bonn 1984) | 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 triple coverings of a smooth algebraic surface, which are non-Galois and satisfy certain genericity condition specified in the text. Such triple coverings can be constructed as the normalization in a cubic extension given by \(X^ 3+aX+b=0\), where \(a\) and \(b\) are rational functions on the base surface \(\Sigma\). It is proved that such normalization has only rational triple points and the numerical characters of its smooth model are computed. In particular, if \(\Sigma\) is taken to be rational ruled, then it gives a surface with a pencil of trigonal curves, which satisfies \(c^ 2_ 1=2p_ g-4\). That is, \(c^ 2_ 1\) is the smallest when \(p_ g\) is fixed. This falls into the class studied by the reviewer [ cf. Ann. Math., II. Ser. 104, 357-387 (1976; Zbl 0339.14024)]. triple coverings Tokunaga, H.: On the construction of triple coverings of algebraic surfaces of certain type. Preprint Special surfaces, Coverings in algebraic geometry Construction of triple coverings of a certain type of algebraic 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 \(L\) be a lattice of \(\mathbb{C}\), \((E,q) = (\mathbb{C}/L,0)\) the corresponding marked elliptic curve and \(z\) a local coordinate of \(E\) at \(q\). We say that a finite ramified marked covering \(\pi : (\Gamma,p) \to (E,q)\) \((\pi (p) = q)\) is ``tangential'' iff all KP solutions associated to the data \((\Gamma, p, \pi (z))\) are elliptic solitons (i.e: \(L\)-periodic in \(x)\). Let us denote \(W (\Gamma)\) the (Mayer-Mumford compactified) jacobian of the latter curve \(\Gamma \), and for any \(n \geq 1\), \(E^{(n)}\) the \(n\)-th symmetric power of \(E\). By intersecting the theta divisor of \(W(\Gamma)\) with the \(E\)-Orbit of any \({\mathcal L} \in W (\Gamma)\), we construct a finite covering \(I : W (\Gamma) \to E^{(n)}\) \((n = \deg \pi)\). The latter morphism was developed and studied by the author and \textit{I.-L. Verdier}, in relationship with the moduli space of elliptic solitons of degree \(n\) over \(E\). The main purpose of this article instead, is to generalise some classical results about (jacobians of) smooth curves to all tangential covers.
We first prove that, whenever \(\pi : (\Gamma,p) \to (E,q)\) is a tangential cover of arithmetic genus \(= n = \deg \pi\), the theta divisor of \(W (\Gamma)\) is the pull-back by \(I\) of \(q + E^{(n - 1)}\). It then follows that the theta divisor of \(W (\Gamma)\) is ample, even if \(\Gamma\) is singular. We finally prove a Torelli-like theorem for compactified jacobians of tangential covers, within the frame of ramified coverings of \(E^{(n)}\). ample divisor; tangential ramified marked covering; marked elliptic curve; elliptic solitons; theta divisor; Torelli-like theorem for compactified jacobians of tangential covers Armando Treibich, Compactified Jacobians of tangential covers, Integrable systems (Luminy, 1991) Progr. Math., vol. 115, Birkhäuser Boston, Boston, MA, 1993, pp. 39 -- 59. Jacobians, Prym varieties, Theta functions and curves; Schottky problem, Coverings in algebraic geometry, Theta functions and abelian varieties Compactified Jacobians of tangential 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 From the summary: ``We verify that universal classes in the cohomology of \(GL_N\) determine
explicit cohomology classes of Frobenius kernels \(G_{(r)}\) of various linear algebraic groups
\(G\). We consider the relationship of \(\varprojlim_r H^*(U_{(r)},k)\) to the rational
cohomology \(H^*(U,k)\) of many unipotent algebraic groups \(U\).''
For instance, let \(T\) be a maximal torus in a simple algebraic algebraic group \(G\) and let
\(U\subset G\) be a connected unipotent subgroup normalised by \(T\).
Then the map \(\varprojlim_r H^*(U_{(r)},k)\to H^*(U,k)\) is shown to be an isomorphism.
``The second half of this paper investigates in detail the cohomology of Frobenius kernels
\((U_3)_{(r)}\) of the Heisenberg group \(U_3 \subset GL_3\).''
The group \(U_3\) is the group of 3 by 3 unipotent upper triangular matrices.
The author constructs a map from ``an integrally closed domain with known generators and
relations'' to \(H^*((U_3)_{(r)},k)\)
``which is a.) injective, b.) surjective onto p-th powers, and c.) has associated graded map
which is both injective and surjective onto p-th powers.''
This construction involves both the Andersen-Jantzen spectral sequence and a
Lyndon-Hochschild-Serre spectral sequence.
``A key ingredient in this construction is the action of the Steenrod algebra on the LHS spectral
sequence which enables us to identify permanent cycles.''
The author anticipates that there also is a suitable action of the Steenrod algebra on the
Andersen-Jantzen spectral sequence. rational cohomology; Frobenius kernels; unipotent algebraic groups Formal groups, \(p\)-divisible groups, Representation theory for linear algebraic groups, Modular representations and characters, Cohomology theory for linear algebraic groups Cohomology of unipotent 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 gonality of a complex projective curve \(C\) is the least degree of a branched covering \(C'\to \mathbb{P}^1\), being \(C'\) the normalization of \(C\). Gonality equal to one is equivalent to rationality of \(C\) and in this sense the gonality is a sort of measure of how far is \(C\) from being rational. This invariant extends to higher dimension to the \textit{degree of irrationality} of an irreducible complex variety \(X\), defined as the minimal degree of a rational covering \(X\) onto \(\mathbb{P}^n\). Further measures of irrationality can be studied as the \textit{connecting gonality} of \(X\), the minimum gonality of an irreducible curve connecting general points \(x,y \in X\) (equal to one if and only if \(X\) is rationally connected); or the \textit{covering gonality}, the minimum gonality of an irreducible curve through a general \(x \in X\) (equal to one if and only if \(X\) is uniruled). First result of the paper under review concerns to covering gonality of smooth degree \(d\) hypersurfaces \(X_d \subset \mathbb{P} ^{n+1}\) (see Thm. A): if \(d \geq n+2\) then the covering gonality of \(X_d\) is greater than or equal to \(d-n\). This is generalized in Cor. B to smooth divisors \(X_d \in |dA|\), \(A\) a very ample divisor on a smooth projective variety. The second result of this paper (see Thm. C) is a proof of the main conjecture of [\textit{F. Bastianelli} et al., J. Algebr. Geom. 23, 313--339 (2014; Zbl 1317.14029)]: under the conditions of Thm. A, if \(X_d\) is very general and \(d\geq 2n+1\) then the irrational degree of \(X_d\) equals \(d-1\); moreover, if \(d\geq 2n+2\) any rational mapping onto \(\mathbb{P}^n\) of degree \(d-1\) is given by a projection from a point of \(X\).
Some interesting open problems are presented in Section 4 and some known results on varieties with many highly secant lines are reproved in Appendix A. hypersurfaces; degree of irrationality; covering gonality; positivity Francesco Bastianelli, Pietro De Poi, Lawrence Ein, Robert Lazarsfeld & Brooke Ullery, ``Measures of irrationality for hypersurfaces of large degree'', Compos. Math.153 (2017), p. 2368-2393 Hypersurfaces and algebraic geometry, Rational and birational maps, Coverings in algebraic geometry Measures of irrationality for hypersurfaces of large degree | 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 \(T\) be a complex torus in \(\mathbb{C}\mathbb{P}_2\) and \(X= \mathbb{C}\mathbb{P}_1 \times T\). By embedding \(X\) into \( \mathbb{C}\mathbb{P}_5\) by the usual Segre map \( \mathbb{C}\mathbb{P}_1 \times \mathbb{C}\mathbb{P}_2 \rightarrow \mathbb{C}\mathbb{P}_5 \) a generic projection \( f: X \rightarrow \mathbb{C}\mathbb{P}_2 \) is obtained by projecting \(X\) from a general plane in \( \mathbb{C}\mathbb{P}_5 \setminus X \) to \(\mathbb{C}\mathbb{P}_2 \). The Galois cover of \(X\) is the closure of the \(n\)-fibred product \(X_{\text{gal}} = \overline{ X \times_{f} \ldots \times_{f} X - \Delta} \) where \( n = \text{deg}(f) \) and \( \Delta \) is the generalized diagonal.
In this paper the authors compute \( \pi_1( X_{\text{gal}}) = \mathbb{Z}^{10}\) stated as theorem 9.3, using braid monodromy techniques developed by \textit{B. Moishezon} and \textit{M. Teicher} [Invent. Math. 89, 601--643 (1987; Zbl 0627.14019)], the van Kampen theorem and various computational methods of groups. generic projection; Seiberg-Witten invariants; fundamental group Meirav Amram, David Goldberg, Mina Teicher, and Uzi Vishne, The fundamental group of a Galois cover of \?\roman\?\textonesuperior \times \?, Algebr. Geom. Topol. 2 (2002), 403 -- 432. Homotopy theory and fundamental groups in algebraic geometry, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Separable extensions, Galois theory, Braid groups; Artin groups, Coverings in algebraic geometry, Computational aspects of algebraic surfaces, Projective techniques in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) The fundamental group of a Galois cover of \(\mathbb{C} \mathbb{P}^1 \times T\) | 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 computes the irregularity of cyclic multiple planes associated to the branch curve with arbitrary singularities. The paper is very well written and has numerous examples and applications.
In more detail, let \(B=\{f(x,y)=0\}\) be a (complex, reduced) plane curve, transversal to the infinite line. The corresponding surface \(S_0=\{z^n=f(x,y)\}\) is in general singular, let \(S\to S_0\) be its resolution. The irregularity \(q(S):=h^1(\mathcal{O}_S)=h^0(\Omega^1_S)\) was first computed by Zariski, in the case when \(B\) has nodes and ordinary cusps only. The irregularity is expressed via \(h^1(\mathbb{P}^2,I_Z)\), where \(Z\) is the support of cusps.
Generalizations and applications of Zariski's formula were studied by many people (Esnault, Libgober, Artal-Bartolo). The author generalizes the formula to the case of \(B\) with arbitrary singularities. The formula is in terms of multiplier ideals of \(B\) and the corresponding jumping numbers..
The proof follows Zariski's ideas. The resolution \(S\to S_0\to\mathbb{P}^2\) is expressed as a standard cyclic covering. Then the theory of cyclic coverings [\textit{R. Pardini}, J. Reine Angew. Math. 417, 191--213 (1991; Zbl 0721.14009)] is applied to express the irregularity. Finally the Kawamata-Viehweg-Nadel vanishing theorem is used.
In \S2 (preliminaries) the cyclic covers and multiplier ideals are nicely introduced. In \S3 the main theorem is proved. \S4 contains examples and applications. In particular, the author uses the main theorem to reconstruct the examples of Zariski pairs (by Artal-Bartolo). In Appendix the author computes in details the case when \(B\) has singularities of the form \(x^m+y^n=0\). cyclic coverings; irregularity; Zariski pairs Naie D.: Irregularity of cyclic multiple planes after Zariski. L'enseignement mathématique 53, 265--305 (2008) Special surfaces, Hypersurfaces and algebraic geometry, Singularities in algebraic geometry, Coverings in algebraic geometry, Ramification problems in algebraic geometry, Vanishing theorems in algebraic geometry The irregularity of cyclic multiple planes after Zariski | 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 \(\Omega =\Sigma P_ j d \log f_ j\) denote a connection composed of endomorphisms \(P_ j\) of a complex vector space and of linear functions \(f_ j\) on \({\mathbb{C}}^ n\). The solutions of the system of differential equations \(dY+\Omega Y=0\) define a local system \({\mathcal L}\) on the complement X of the affine hyperplanes \(A_ j\) defined by \(f_ j\). The homology \(H_ k(X, {\mathcal L})\) vanishes for \(k\neq n\) if the eigenvalues of \(P_ j\) and certain sums \(P_{j_ 1}+...+P_{j_ q}\) are not entire. Furthermore for real \(f_ j's\) a base of \(H_ n(X, {\mathcal L})\) is given in terms of the relative compact chambers of the real part of X. The proof is rather short: After a monoidal transformation along \(A_{j_ 1}\cap...\cap A_{j_ q}\) the Picard-Lefschetz formulas are applied. complex differential equations; homology with local coefficients; complements of hyperplanes; monoidal transformation; Picard-Lefschetz formulas koh T. Kohno, Homology of a local system on the complement of hyperplanes. Proc. Japan Acad. Ser. A Math. Sci. \textbf 62 (1986), 144-147. Homology with local coefficients, equivariant cohomology, Ordinary differential equations in the complex domain, Analytic sheaves and cohomology groups, Coverings in algebraic geometry, Classical real and complex (co)homology in algebraic geometry Homology of a local system on the complement of hyperplanes | 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 is devoted to Klein's resolvent problem for algebraic functions: Given an algebraic function \(\mathbf{z}\) on an irreducible variety \(X\), what is the smallest number \(k\) such that the function \(\mathbf{z}\) can be rationally induced from an algebraic function \(\mathbf{w}\) on some variety \(Y\) of dimension \(\leq k\)? The author completely solves the problem for algebraic functions unramified on \((\mathbb{C}\setminus\{ 0\} )^{n}\) using geometric and topological methods. In the first part of the paper, coverings over topological \(n\)-tori \((S^{1})^{n}\) are studied. The author introduces the notion of the topological essential dimension of a covering and then proves that the topological essential dimension of a covering over \(n\)-torus is equal to the rank of its monodromy group. In the second part of the paper these topological results are put in an algebraic context and used to prove that the algebraic essential dimension of an algebraic function unramified over the algebraic \(n\)-torus \((\mathbb{C}\setminus\{ 0\} )^{n}\) is equal to the rank of its monodromy group. This provides the solution of the starting problem. Furthermore, the author proves that the algebraic essential dimension of the universal algebraic function \(\mathbf{z}\) defined by the equation \(\mathbf{z}^{n}+x_{1}\mathbf{z}^{n-1}+\cdots+x_{n}=0\) is at least \(\lfloor n/ 2\;\rfloor\). This paper is a nice example of how results obtained by methods and techniques of one mathematical branch can be successfully applied to a problem of another mathematical branch, which is fruitful for both of them. Coverings in algebraic geometry, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Homotopy groups Coverings over tori and topological approach to Klein's resolvent problem | 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 \(\mathbb{C}\) be the field of complex numbers, let \(\mathbb{C} [x]\) be the ring of polynomials in an indeterminate \(x\) over \(\mathbb{C}\), and let \(t\) be another indeterminate. The monodromy group of an element \(f(x)\) of \(\mathbb{C} [x]\) is the Galois group of the polynomial \(f(x) - t\) over \(\mathbb{C} (t)\). It may be described also as the fundamental group of the Riemann surface for the function \(f(z)\) defined on the extended complex plane. The author develops and applies to the classification of finite simple groups a process of elimination to determine what groups may be the monodromy group of a polynomial \(f(x)\) which is indecomposable (i.e. there do not exist nonlinear polynomials \(g(x)\) and \(h(x)\) in \(\mathbb{C} [x]\) such that \(f(x) = g(h(x)))\). The criteria for exclusion do not appear always to be fully stated. For example, the reviewer was unable to understand the argument for eliminating the subgroup of \(P \Gamma L_2 (9)\) of type (2,2,2;10), which is isomorphic to the subgroup of the symmetric group \(S_{10}\) generated by the permutations (3,10)(5,9)(7,8), (1,10)(3,9)(5,8), and (1,2)(3,4)(5,6).
By further refinement of the process of elimination, the author determines the possible monodromy groups for polynomials with rational coefficients. In an appendix to the paper, he uses monodromy groups to prove results obtained by \textit{J. F. Ritt} [Trans. Am. Math. Soc. 23, 51-66 (1922; JFM 48.0079.01)] about decomposition of polynomials. fundamental group of a Riemann surface; indecomposable polynomial; monodromy group Müller, P, Primitive monodromy groups of polynomials, Contemp. Math., 186, 385-401, (1995) Polynomials in real and complex fields: factorization, Separable extensions, Galois theory, Coverings in algebraic geometry, Finite simple groups and their classification Primitive monodromy groups of 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 Report on two works of \textit{S. Lang} [ Ann. Math. (2) 64, 285--325 (1956; Zbl 0089.26201)] and [Bull. Soc. Math. Fr. 84, 385--407 (1956; Zbl 0089.26301)]. Class field theory, Geometric class field theory, Coverings in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Class field theory for nonramified coverings of algebraic varieties (after S. Lang) | 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 smooth projective surface \(X\) is said to have the infinitesimal Torelli property if the natural map induced by cup product \(H^1(T_X)\to \Hom(H^0(\Omega^1_X), H^0(\omega_X))\) is injective. In this paper the author shows that infinitesimal Torelli holds for a double cover \(X\) of a smooth projective surface \(S\) branched on a bicanonical curve, if \(S\) satisfies the following assumptions: 1) \(h^1({\mathcal O}_S)=0\), \(h^0(\omega_S)\geq 2\) and \(K^2_S\geq 5\); 2) the canonical system \(| K_X| \) contains a reduced irreducible divisor. Infinitesimal Torelli; surface of general type; double cover. Surfaces of general type, Coverings in algebraic geometry, Torelli problem Infinitesimal Torelli theorem for double coverings of surfaces of general 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 In: Number Theory, Algebraic Geometry, Commucative Algebra, in Honor Y. Akizuki, 147-167 (1973; Zbl 0271.14015), \textit{S. Iitaka} made the conjecture
\(U_n\): If \(V\) is a smooth projective variety with \({\mathbb{C}}^n\) as its universal cover, then a finite unramified cover of \(V\) is an abelian variety (such \(V\) will be called para-abelian variety).
In an early version of this paper, the author proved that \(U_3\) is true using \(\partial\)-étale cohomology theory developed in his early paper [``Global structure of an elliptic fibration'' (Kyoto Univ. 1996)]. Here the author generalizes and proves, among other things, the following:
(1) If \(V\) is smooth projective of type \(U\) (this is satisfied if \(V\) has \({\mathbb{C}}^n\) as its universal cover) and if the canonical divisor is semi-ample (i.e., a positive multiple of it is base point free), then \(V\) is a para-abelian variety.
(2) If \(V\) is smooth projective of type \(U\) and if the canonical divisor \(K_F\) of a general fibre \(F\) of the Albanese mapping \(\alpha : V \rightarrow \text{Alb}(V)\) is semi-ample, then \(V\) is a para-abelian variety and \(\alpha\) is an étale fibre bundle. universal covering; para-abelian variety; canonical divisor; Albanese mapping N. Nakayama: Projectiva algebraic varieties whose universal covering spaces are biholomorphic to \(\mathbb C^n\), J. Math. Soc. Japan. 51 (1999), 643--654. Coverings in algebraic geometry, Abelian varieties and schemes, Minimal model program (Mori theory, extremal rays), Families, moduli, classification: algebraic theory Projective algebraic varieties whose universal covering spaces are biholomorphic to \(\mathbb{C}^n\) | 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 The authors prove that the category of perverse sheaves on the complex two-space \({\mathbb{C}}^ 2\), constructible with respect to the stratification \(\{0\}\subset \{y^ n=x^ m\}\subset {\mathbb{C}}^ 2\), \(n\leq m\), is equivalent to the category of \((n+2)\)-tuples of vector spaces \(A,B_ 1,...,B_ n,C\) together with maps \(A\rightleftarrows^{q_ k}_{p_ k}B_ k\rightleftarrows^{s_ k}_{r_ k}C\) and \(\theta_ k:B_ k\to B_{k+m}\) (all the indices are to be considered as integers modulo n), which satisfy some relations, explicitely given. The proof uses methods of the authors given in C. R. Acad. Sci., Paris, Ser. I. 299, 443-446 (1984; Zbl 0581.14013), and Invent. Math. 84, 403-435 (1986; Zbl 0597.18005). As an application one classifies perverse sheaves with no vanishing cycles at the origin for the case \(y^ 2=x^ 3\); in this case the nontrivial irreducible perverse sheaves are parameterized by one complex number. fundamental group; perverse sheaves; stratification MacPherson, R. D.; Vilonen, K., Perverse sheaves with singularities along the curve \textit{y}\textit{n}\ =\ \textit{x}\textit{m}, \textit{Comment. Math. Helv.}, 63, 89-102, (1988) Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Analytic subsets of affine space, Coverings in algebraic geometry Perverse sheaves with singularities along the curve \(y^ n=x^ m\) | 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 After a historical discussion of classical uniformisation results for Riemann surfaces, of problems appearing in higher dimensions, and of uniformisation results for projective manifolds with trivial or ample canonical bundle, we introduce the basic technical concepts and sketch the ideas of the proofs for recent uniformisation theorems for singular varieties obtained by the authors in collaboration with Thomas Peternell. Uniformization of complex manifolds, Rational and birational maps, Notions of stability for complex manifolds, Coverings in algebraic geometry, Minimal model program (Mori theory, extremal rays), Projective connections, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), (Equivariant) Chow groups and rings; motives, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Uniformisation of higher-dimensional minimal 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 Let S be a complex projective nonsingular algebraic surface of general type with \(p_ g=0\), \(K^ 2=1\). Then, as is known, the torsion group \(Tors=Tors H_ 1(S,{\mathbb{Z}})\) is \({\mathbb{Z}}_ n\), \(1\leq n\leq 5\). A description of the corresponding moduli space for \(3\leq n\leq 5\) has been given by \textit{M. Reid} [J. Fac. Sci., Univ. Tokyo, Sect. I A 25, 75-92 (1978; Zbl 0399.14025)]. The remaining cases \(Tors=\{1\}\), \({\mathbb{Z}}_ 2\) also do occur, as appropriate examples show. In the paper under review the author constructs a 4 parameter family of surfaces as above with \(\pi_ 1={\mathbb{Z}}_ 2\). This is interesting since the only other example known with \(Tors={\mathbb{Z}}_ 2\), due to \textit{F. Oort} and \textit{C. Peters} [Indagationes Math. 43, 399-407 (1981; Zbl 0523.14025)] seems hard to deform. The surfaces are obtained by resolving the nodes of quotients Y/G, where Y is a complete intersection of 4 quadrics of \({\mathbb{P}}^ 6\) and G is a group of order 16. In particular, since G has a subgroup \({\mathbb{Z}}_ 8\) acting freely, Y/G is double covered by \(Y/{\mathbb{Z}}_ 8\), which is a Godeaux surface with \(p_ g=0\), \(K^ 2=2\), \(\pi_ 1={\mathbb{Z}}_ 8\). The author produces also two similar constructions of 4 parameter families of surfaces with \(\pi_ 1={\mathbb{Z}}_ 4\), double covered by some Godeaux-Reid surfaces. double cover; group action; fundamental group; surface of general type; torsion group; Godeaux-Reid surfaces Barlow, R, Some new surfaces with \(p_g=0\), Duke Math. J., 51, 889-904, (1984) Special surfaces, Families, moduli, classification: algebraic theory, Group actions on varieties or schemes (quotients), Coverings in algebraic geometry Some new surfaces with \(p_ g=0\) | 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 \(k[\Delta]\) be the Stanley-Reisner ring for the simplicial complex \(\Delta\) over the field \(k\). The first section gives a necessary and sufficient condition for \(k[\Delta]\) to have a linear resolution, under the assumption that \(k[\Delta]\) is a Buchsbaum ring. The condition is given in terms of the reduced homology of \(\Delta\) and the \(a\)-invariants of its links. This condition improves a result of \textit{J. Hibi} [J. Algebra 179, 127--136 (1996; Zbl 0839.55012 )]. The second section is concerned with \((d-1)\)-dimensional simplicial complexes with initial degree \(d\). (Once the vertex set \(V\) is fixed, then every \((d-1)\)-element subset of \(V\) is a simplex of \(\Delta\), some \(d\)-element subsets of \(V\) are faces of \(\Delta\), and \(\Delta\) does not contain any larger simplices.) Every such \(\Delta\) has a Cohen-Macaulay cover \(\widetilde{\Delta}\). The \(d-1\) reduced homology, \(\widetilde{H}_{d-1}\), of the two simplicial complexes have the same dimension and \(\widetilde{\Delta}\) is obtained from \(\Delta\) by adjoining \(\alpha\) \(d\)-element subsets of \(V\) to \(\Delta\), where \(\alpha=\text{dim}_k \widetilde{H}_{d-2}(\Delta,k)\). If \(\Delta\) is a Buchsbaum complex, then there is a chain of Buchsbaum complexes:
\[
\Delta=\Delta_{\alpha}\subset \Delta_{\alpha-1}\subset \dots \subset \Delta_0=\widetilde{\Delta},
\]
all with the same \(\text{dim}_k\widetilde{H}_{d-1}\); but \(\text{dim}_k \widetilde{H}_{d-2}\) drops by one at each step. (When \(d=2\) this process amounts to adding edges to a forest in order to make a tree.) As an application the authors determine the \(h\)-vectors of the \(3\)-dimensional Buchsbaum Stanley-Reisner rings with initial degree \(3\). Alexander duality; Buchsbaum ring; linear resolution; multiplicity; regularity Terai, N.; Yoshida, K.: Buchsbaum Stanley--Reisner rings and Cohen--Macaulay covers. Comm. algebra 34, 2673-2681 (2006) Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Coverings in algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Syzygies, resolutions, complexes and commutative rings Buchsbaum Stanley--Reisner rings and Cohen--Macaulay 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 Let \(Y\) be a simply connected open neighbourhood of \(0\in \mathbb{C}^n\) and \(\pi:X\to Y\) be a finite Galois covering. Let \(B_\pi= \sum^\ell_{v= 1} r_vB_v\), \(B_1,\dots,B_\ell\) be the irreducible components of the branch locus of \(\pi\) and \(r_1,\dots,r_\ell\) the corresponding ramification indices of \(\pi\) along \(B_j\).
Galois coverings \(\pi:X\to Y\) are studied such that \(B_\pi=D\) for a given divisor \(D\) on \(Y\) and \(\text{Gal} (X\mid Y)= \{g\in \Aut (X) \mid\pi \circ g=\pi\}\).
[For part I of the paper see the author, J. Math. Soc. Japan 51, 35-44 (1999; Zbl 0933.32040)]. ramification; Galois coverings Modifications; resolution of singularities (complex-analytic aspects), Coverings in algebraic geometry, Coverings of curves, fundamental group Galois covering singularities. II | 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 prove a generalization of \textit{A. Grothendieck}'s Lefschetz theorem for complete intersections [Séminaire de géométrie algébrique: Cohomologie locale des faisceaux cohérents et théoremes de Lefschetz locaux et globaux (1962; Zbl 0159.50402); XII 3.5]. \(\pi_1 (X)\) will denote the algebraic fundamental group of a scheme \(X\). If \(k\) is a field, \(\overline k\) will denote an algebraic closure of \(k\). Our result is:
Theorem 1. Suppose that \(k\) is a field, \(W\) is a smooth, geometrically connected subvariety of \(\mathbb{P}^m_k\) of dimension \(n\) and \(Z \subset W\) is a closed subscheme set theoretically defined by the vanishing of \(r\) forms of \(\mathbb{P}^m_k\) on \(W\).
1. If \(r \leq n-1\) then \(Z\) is geometrically connected and there is a surjection \(\pi_1(Z) \to \pi_1(W)\).
2. If \(r \leq n-2\), then \(\pi_1(Z) \cong \pi_1(W)\).
Corollary 2. Suppose that \(k\) is a field and \(Z \subset \mathbb{P}^n_k\) is a closed subscheme set theoretically defined by \(r\) forms.
1. If \(r\leq n-1\) then \(Z\) is geometrically connected.
2. If \(r\leq n-2\), then \(\pi_1(Z) \cong \text{Gal} (\overline k/k)\) where \(\overline k\) is an algebraic closure of \(k\).
The corresponding theorem for the topological fundamental group of a complex projective variety follows from a paper by \textit{H. A. Hamm} [in: Singularities, Summer Inst., Arcata 1981, Proc. Symp. Pure Math. 40, Part 1, 547-557 (1983; Zbl 0525.14011)] and the theorem of II, 1.2 in the book by \textit{M. Goresky} and \textit{R. MacPherson}, ``Stratified Morse theory'' (1988; Zbl 0639.14012). Their proofs use different methods (Morse theory) and do not extend to positive characteristic. simple connectedness; Lefschetz theorem; algebraic fundamental group Cutkosky, S. D.: Simple connectedness of algebraic varieties, Proc. amer. Math. soc. 125, No. 3, 679-684 (1997) Coverings in algebraic geometry, Topological properties in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry Simple connectedness of projective 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 Let \(\pi: X\rightarrow Y\) be a crepant resolution of an affine symplectic variety \((Y,\sigma)\). The symplectic form \(\sigma\) on \(Y_{reg}\) can be lifted and extended to a symplectic form on \(X\) and yields Poisson structures on \(Y\) and \(X\). The author studies the Poisson deformation of these structures in the situation where \(Y\) and \(X\) are both equipped with a \(\mathbb C^*\)-action compatible with \(\pi\) and where the \(\mathbb C^*\)-action on \(Y\) has a unique fixed point \(0\in Y\) such that the cotangent space \(m_{Y,0}/m^2_{Y,0}\) is decomposed into onedimensional eigenspaces with positive weights. The author defines and considers the Kuranishi spaces PDef\((X)\) and PDef\((Y)\) of Poisson deformations \({\mathcal X}\rightarrow {\mathbb{A}^d}\) of \(X={\mathcal X}_0\) and \({\mathcal Y}\rightarrow{\mathbb{A}^d}\) of \(Y={\mathcal Y}_0\) which are both universal at \(0\in {\mathbb{A}^d}\). In a previous paper [Duke Math. J. 156, No. 1, 51--85 (2011; Zbl 1208.14028)] he proved that PDef\((X)\) and PDef\((Y)\) are smooth varieties with the same dimension and that the natural map \(f:\) PDef\((X)\rightarrow \) PDef\((Y)\) induced by \(\pi\) is a finite branched covering. The main result of the present paper states that \(f\) is a Galois covering. The Galois group is a product of Weyl groups of specific root systems related to the singularity type of \(Y\). These groups are calculated in detail. An analogous result for a projective symplectic variety \(Y\) was proved by \textit{E. Markmann} [\url{arXiv: 0807.3502}]. The methods used in the present paper yield a new proof of Markmann's result. As an application of the main result the author constructs explicitly the universal Poisson deformation of the normalization \(\tilde{\mathcal O}\) of a nilpotent orbit closure \(\overline{\mathcal O}\) in a complex simple Lie algebra when \(\tilde{\mathcal O}\) has a crepant resolution. Poisson deformation; Kuranishi space; Galois covering; nilpotent orbit closure; crepant resolution Namikawa, Y., \textit{Poisson deformations of affine symplectic varieties, II}, Kyoto J. Math., 50, 727-752, (2010) Moduli, classification: analytic theory; relations with modular forms, Deformations of complex structures, Coverings in algebraic geometry, Compact complex \(n\)-folds, Compact Kähler manifolds: generalizations, classification, Local deformation theory, Artin approximation, etc. Poisson deformations of affine symplectic varieties. II | 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 \(X\) be an irreducible projective variety of dimension \(n\) over an algebraically closed field \(k\), and let \(f:X\to\mathbb{P}^n\) be a finite morphism. Denote by \(d\) the geometric degree of \(f\), i.e., the separable degree of the extension \(k(X)/k(\mathbb{P}^n)\) of function fields. For each \(x\in X\), one can define the local degree \(e_f(x)\) of \(f\) at \(x\), which may be thought of as the number of sheets of the covering \(X\to\mathbb{P}^n\) that come together at \(x\). When \(X\) is a non-singular complex variety, \(e_f(x)\) coincides with the usual topological local degree. Our main result generalizes the classical fact
that any non-trivial irreducible covering of \(\mathbb{P}^n\) ramifies:
Theorem 1. There exists at least one point \(x\in X\) at which \(e_f(x) \geq\min(d,n + 1)\).
As an application,
we deduce:
Theorem 2. If \(X\) is normal, and admits a branched covering \(X\to\mathbb{P}^n\) of geometric degree \(s_n\), then \(X\) is algebraically simply connected.
The basic tool used in the proof of Theorem 1 is the recent connectedness theorem of \textit{W. Fulton} and \textit{J. Hansen} [Ann. Math. (2) 110, 159--166 (1979; Zbl 0389.14002)]. ramification; geometric degree; local degree; non-singular complex variety; branched covering; simply connected; connectedness theorem Terence Gaffney and Robert Lazarsfeld, On the ramification of branched coverings of \?\(^{n}\), Invent. Math. 59 (1980), no. 1, 53 -- 58. Coverings in algebraic geometry, Topological properties in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry On the ramification of branched coverings of \(\mathbb{P}^n\) | 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 \(X\) be a scheme defined over an algebraically closed field \(k\) of characteristic \(p\) and let \(\mathcal D_X\) be the sheaf of \(k\)-linear differential operators on \(X\). The scheme \(X\) is called \(D\)-affine if every \(\mathcal D_X\)-module \(M\) is generated over \(\mathcal D_X\) by its global sections and \(H^i(X,M) = 0\) for all \(i > 0\). First, the author recalls some well known facts. For example, every flag variety in characteristic zero is \(D\)-affine (see [\textit{A. Beilinson} and \textit{J. Bernstein}, C. R. Acad. Sci., Paris, Sér. I 292, 15--18 (1981; Zbl 0476.14019)]). However, in positive characteristic it is not true (see [\textit{M. Kashiwara} and \textit{N. Lauritzen}, C. R., Math., Acad. Sci. Paris 335, No. 12, 993--996 (2002; Zbl 1016.14009)]), although some flag varieties are still \(D\)-affine. Next, any smooth \(D\)-affine projective toric variety is a product of projective spaces (see [\textit{J. F. Thomsen}, Bull. Lond. Math. Soc. 29, No. 3, 317--321 (1997; Zbl 0881.14020)]), etc. In the paper under review the author describes some further results concerning the classification of smooth projective \(D\)-affine varieties over fields of any characteristic. In particular, he proves that any smooth projective \(D\)-affine variety is algebraically simply connected and its image under a fibration is \(D\)-affine as well. In zero characteristic such \(D\)-affine varieties are, in fact, uniruled. Moreover, assuming \(p=0\) or \(p>7\), he shows that a smooth projective surface is \(D\)-affine if and only if it is isomorphic to either \(\mathbb P^2\) or \(\mathbb P^1\times \mathbb P^1\). Some results are also obtained for three-folds \(D\)-affine varieties. In positive characteristic case the author applies his own modified version of the generic semipositivity theorem due to \textit{Y. Miyaoka} [Proc. Symp. Pure Math. 46, No. 1, 245--268 (1987; Zbl 0659.14008)]. smooth projective varieties; flag varieties; Miyaoka's semipositivity theorem; cotangent bundle; rational surfaces; divisorial contractions; fibrations; crystalline differential operators; étale fundamental group; semistability; reflexives sheaves; semipositive sheaves; uniruled varieties; Riemann-Hilbert correspondence; stable Higgs bundle; Chern classes; flat connections; Artin's criterion of contractibility; Kodaira dimension; Hirzebruch surface; canonical divisor; surfaces of general type; Barlow's surfaces; del Pezzo surfaces; Fano three-folds Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules On smooth projective D-affine 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 Let \(V\) be a smooth projective surface over a number field \(k\), and suppose that there are two elliptic fibrations \(f_i:V\rightarrow C_i\) defined over \(k\), such that no fiber of \(f_1\) is algebraically equivalent to any fiber of \(f_2\). In practice the most interesting cases will be \(K3\) surfaces and Enriques surfaces. The author defines the notion of a twist \(W\) of \(V\), with an associated isomorphism \(\varphi:W_{\overline k}\rightarrow V_{\overline k}\).
The first main result then says that for any \(d\) there is an explicitly computable Zariski-closed \(Z\), properly contained in \(V\), such that for any extension \(K/k\) with \([K:\mathbb{Q}]\leq d\), and for any twist \(W\) of \(V\), the set \(W(K)\) is Zariski dense in \(W\) as soon as it contains a point outside \(\phi^{-1}(Z)\).
This theorem was motivated by a result of Swinnerton-Dyer (to appear), but is stronger, in that it is uniform in all twists and all extensions \(K\) of bounded degree. The paper goes on to prove a second theorem, which has a uniformity over extensions \(K\) of arbitrary degree, and requires that \(W\) should contain sufficiently many points outside \(\phi^{-1}(Z)\).
The author conjectures that the first result might hold for \(K3\) surfaces even with \(Z=\emptyset\). In particular he conjectures that every \(a\in\mathbb{Q}\) is representable as
\[
\frac{x^4-y^4}{z^4-w^4}=a.
\]
\(K3\) surfaces; rational points; Zariski dense F. Izadi, K. Nabardi, A note on diophantine equation \(A^4+D^4=2(B^4+C^4)\) Rational points, \(K3\) surfaces and Enriques surfaces, Elliptic surfaces, elliptic or Calabi-Yau fibrations Density of rational points on elliptic 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 Given a curve \(C\) of genus \(2\) defined over a field \(k\) of characteristic different from \(2\), with a Jacobian \(J\), we show that the two-coverings corresponding to elements of a large subgroup of \(\mathrm{H}^1(\mathrm{Gal}(k^s/k),J[2](k^s))\) (containing the Selmer group when \(k\) is a global field) can be embedded as an intersection of \(72\) quadrics in \(\mathbb{P}^{15}_k\), just as the Jacobian \(J\) itself. Moreover, we actually give explicit equations for the models of these twists in the generic case, extending the work of \textit{D. M. Gordan} and \textit{D. Grant} [Trans. Am. Math. Soc. 337, No. 2, 807--824 (1993; Zbl 0790.14028)] which applied only in the case when all Weierstrass points are rationnal. In addition, we describe elegant equations of the Jacobian itself and answer a question of Cassels and Flynn concerning a map from the Kummer surface in \(\mathbb{P}^3\) to the desingularized Kummer surface in \(\mathbb{P}^5\). genus 2; explicit equations; Selmer group; two-covering; Jacobian; Kummer surface Flynn, E. Victor; Testa, Damiano; Van Luijk, Ronald: Two-coverings of Jacobians of curves of genus 2. Proc. lond. Math. soc. (3) 104, No. 2, 387-429 (2012) Coverings of curves, fundamental group, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Abelian varieties of dimension \(> 1\), Jacobians, Prym varieties Two-coverings of Jacobians of curves of genus 2 | 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 problem of separating two semialgebraic sets in an algebraic \(\mathbb{R}\)-variety is proved to be decidable in dimension 2 with geometric and topologic means. An interpretation of the equivalence between the known criteria of separation is also given. decidability; separating semialgebraic sets Semialgebraic sets and related spaces, Decidability of theories and sets of sentences Decidability of the separation problem in dimension 2 | 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 old problem in Algebraic Geometry asks about the maximum number \(\mu(n)\) of isolated singular points on a projective surface \(F\) of degree \(n\), contained in \(\mathbb{P}^3=\mathbb{P}^3(\mathbb{C})\). The paper under review is a survey on this subject, written by one of its a foremost experts.
Although its statement is elementary, this problem is very hard. The value of \(\mu(n)\) has been found for \(n<6\). For \(n>6\) only upper bounds for the number of isolated singularities were obtained, but probably they (specially for \(n\) large) are not sharp. Many ingenious examples of surfaces of degree \(n>6\) with many nodes were found, but the number is probably \(<\mu(n)\).
The article contains interesting historical information, for instance on a failed attempt by Severi to get a simple bound, based on the study of certain moduli, which turned out to be incorrect. There is a discussion of the proof of \textit{A. Beauville} [Invent. Math. 55, 121--140 (1979; Zbl 0403.14006)] of the formula \(\mu(5)=31\), using a method that (with some homological arguments) reduces the problem to one on ``binary codes'', i.e., finite dimensional vector spaces over \(\mathbb{Z}_2\). This method, with more labour, yields the formula \(\mu(6)=65\), and might work for higher values of \(n\). The author also explains the work of Miyaoka, who found interesting general bounds and studied the asymptotic behaviour of \(\mu(n)/n^3\).
In addition the author discusses numerous examples of surfaces with many nodes, and several other related topics.
The paper is beautifully written, although some parts are expressed in the language of classical Italian Geometry, which might be a little challenge for some modern readers. The article contains a very extensive bibliography, that ranges from a 1750 article by Cramer to very recent preprints not published yet. surfaces in projetive 3-space; isolated singularities; rational double points Gallarati, D, Superficie algebriche con molti punti singolari isolati, Bull. Math. Soc. Sci. Math. Roumanie, 55, 249-274, (2012) Special surfaces, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Computational aspects of algebraic surfaces, Hypersurfaces and algebraic geometry Algebraic surfaces with many isolated singular points (Superficie algebriche con molti punti singolari isolati) | 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 work deals with the embedded join \(XY\) of two subschemes \(X,Y\) of \(\mathbb{P}^n_k\). The join is again a subscheme of \(\mathbb{P}^n_k\), which as a set consists of the closure of the union of all lines \(\overline{xy}\) through distinct points \(x,y\) of \(X,Y\) (at least if \(k\) is algebraically closed). In the case \(X=Y\) the join construction yields the classical secant variety. Various authors have taken up the subject [for instance, \textit{H. Flenner} and \textit{W. Vogel}, Math. Ann. 302, 489--505 (1995; Zbl 0844.14002)], often emphasizing the dimension of the join and the relation to intersection theory. In the present article instead we aim at a more refined study of the (not necessarily reduced) ideal defining the embedding \(XY\subset\mathbb{P}^n_k\). One of the main tools we use is the deformation to the monomial case. We describe the precise behavior of the ideal of an embedded join under this mechanism (section 2). The punch line here is given by theorems 2.2 and 2.3. This reduction procedure leads us naturally to an exploration of the join of subschemes defined by (not necessarily square-free) monomials (section 3). As an application we estimate the initial degree of the ideals defining embedded joins and higher secant varieties, even in the non-monomial case (section 4). We illustrate our results with a few classical examples of varieties of determinantal type. The list includes generic matrices and generic symmetric and alternating matrices. Certain generalized catalecticant loci are also treated (section 5). secant variety; embedded join Simis, A., Ulrich, B.: On the ideal of an embedded join. J. Algebra 226, 1--14 (2000) Projective techniques in algebraic geometry On the ideal of an embedded join. | 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 in detail the \(F\)-blowups of certain normal surface singularities. The \(F\)-blowup of a variety was introduced by \textit{T. Yasuda} [Am. J. Math. 134, No. 2, 349--378 (2012; Zbl 1251.14002)]. Its interaction to \(F\)-pure and \(F\)-regular singularities has not been fully expored. This article deals with understanding the properties of \(F\)-blowups of non \(F\)-regular rational normal double points and simple elliptic singularities, in relation to normality, smoothness, and stabilization of the \(F\)-blowup sequence. The techniques used combine classical results on normal surface singularities with computations performed with the help of Macaulay2 using two computational tools implemented here. Given a module, the first computes an ideal such that the blowups at the ideal and module coincide, based upon \textit{O. Villamayor U.}'s work [J. Algebra 295, No. 1, 119--140 (2006; Zbl 1087.14011)]. The second tool computes the Frobenius pushforward \(F_*M\) of a given module \(M\). Among other things, in the case of the rational normal double points, the authors exhibit two non-\(F\)-regular such surfaces for which the \(e\)th \(F\)-blowup is the minimal resolution, for \(e \geq 2\). For simple elliptic singularities, the authors determine the structure of \(F\)-blowups up to normalization. Some of the results build upon previous work of \textit{N. Hara} and \textit{T. Sawada} [RIMS Kôkyûroku Bessatsu B24, 121--141 (2011; Zbl 1228.13009)]. \(F\)-blowups; \(F\)-pure surface; \(F\)-regular surface; rational double points; simple elliptic singularities \beginbarticle \bauthor\binitsN. \bsnmHara, \bauthor\binitsT. \bsnmSawada and \bauthor\binitsT. \bsnmYasuda, \batitle\(F\)-blowups of normal surface singularities, \bjtitleAlgebra Number Theory \bvolume7 (\byear2013), page 733-\blpage763. \endbarticle \OrigBibText N. Hara, T. Sawada and T. Yasuda, \(F\)-blowups of normal surface singularities, Algebra Number Theory 7 (2013), 733-763. \endOrigBibText \bptokstructpyb \endbibitem Singularities in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) \(F\)-blowups of normal surface singularities | 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 \(\mathbb{P}^N\) be a projective space over an algebraically closed field of characteristic zero. Let \(X\subset \mathbb{P}^N\) be a closed, irreducible subvariety, not lying on a hyperplane. The \(k\)-th higher secant variety of \(X\), denoted \(X^k\), is the closure of the union of all linear spaces spanned by \(k\) points of \(X\). We prove that \(I(X^k)\), the homogeneous ideal of \(X^k\), is contained in the \(k\)-th symbolic power of \(I(X)\). As a consequence, \(X^k\) lies on no hypersurface of degree less than \(k+1\). Furthermore, if \(X\) is a curve, and \(\deg X^k=k+1\), we prove that \(X\) is a rational normal curve. higher secant variety; homogeneous ideal; symbolic power; rational normal curve Catalano-Johnson, M. L.: The homogeneous ideal of higher secant varieties. J. pure appl. Algebra 158, 123-129 (2001) Varieties of low degree, Projective techniques in algebraic geometry, Plane and space curves The homogeneous ideal of higher secant 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 Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard to handle, it has been shown in a previous paper by the author [J. Algebr. Geom. 11, 245--256 (2002; Zbl 1054.14035)] that there exists a partial resolution of singularities which transforms a bundle of possibly badly singular curves into a bundle of nodal and cuspidal plane cubics. In cases which are of interest for classification theory, the total spaces of these bundles will clearly be projective. It is, however, generally false that an arbitrary bundle of plane cubics is globally projective. For that reason the question of projectivity and the study of moduli seems to be of interest, and the present work gives a characterization of the projective bundles. families of singular rational curves; partial resolution of singularities Fibrations, degenerations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Projective bundles of singular plane cubics | 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 aim of the present paper, which is a continuation of a previous series of works of the both authors, is to determine the structure of a nonsingular \(n\)-dimensional variety \(X\) in \(\mathbb{P}^N_k\), \(n\geq 4\), and \(k\) an algebraically closed field of characteristic 0, which is swept out by its \(m\)-dimensional subvarieties of degree 2 in \(\mathbb{P}^N_k\).
A first result is: If \(nm\) is large enough, then \(X\) admits either a linear or a quadric fibration.
The main result of the paper is the following: If the quadrics in \(X\) all pass through a same point, then \(X\) is either a linear subspace or a nonsingular quadric of \(\mathbb{P}^N_k\).
In the introduction the authors explain in details three motivations for their study and in the course of the paper they also present some pertinent problems. fibration; hyperquadrics Kachi, Y., Sato E.: Segre's Reflexivity and an Inductive Characterization of Hyperquadrics. Mem. Am. Math. Soc., 160, 763 (2002) Families, moduli, classification: algebraic theory, Fibrations, degenerations in algebraic geometry, Projective techniques in algebraic geometry, \(n\)-folds (\(n>4\)), Minimal model program (Mori theory, extremal rays), Rational and birational maps, \(4\)-folds Segre's reflexivity and an inductive characterization of hyperquadrics | 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 0745.00060.]
Soit \(K\) un corps de caractéristique zéro, muni d'une valeur absolue \(| |\). L'étude de la distance d'un point \(x\in K^ n\) à une sous-variété algébrique \(V\) définie sur \(K\) de \(K^ n\) est un des points clefs de l'approche effective du théorème des zéros de Hilbert, tel qu'établi par W. D. Brownawell. Si \(f_ 1,\dots,f_ k\) sont des générateurs de l'idéal de définition de \(V\), S. Ji, J. Kollár et B. Shiffman ont montré
\[
\text{dist}(x,V)^ m\leq C(m, \mathbf{f}),\;\max_ i| f_ i(x)|\;(1+\| x\|)^{D^ \mu}\tag{*}
\]
où \(m\leq D^ \mu\), \(\mu=\min(k,n)\), \(\| \|\) désigne la norme euclidienne et dist la distance euclidienne dans \(K^ n\).
L'A. montre dans ce texte comment contrôler plus précisément l'exposant \(m\). Par exemple, si \(x\) parcoure une sous-variété linéaire \(L\) de \(K^ n\) de codimension égale à \(\dim V\), telle que \(L\cap V\) soit fini alors \(m\) peut-être pris inférieur ou égal à la multiplicité de \(V\) dans la direction \(L\).
Combinant ceci avec la version forte du théorème des zéros il retrouve une versione projective de l'inégalité de Łojasiewicz (*) mentionnée ci-dessus. Łojasiewics inequality; Hilbert Nullstellensatz; Cayley-Chow form Brownawell, W. D.: Distance to common zeros and lower bounds for polynomials, , 51-60 (1992) Diophantine approximation, transcendental number theory, Effectivity, complexity and computational aspects of algebraic geometry, Holomorphic functions of several complex variables Distance to common zeros and lower bounds 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 A smooth projective variety \(X\) (over an uncountable algebraically closed field \(k)\) is said to be rationally connected (R. C. for short) if any two generic points of \(X\) are contained in some connected curve all irreducible components of which are rational (possibly singular). Unirational varieties, as well as Fano varieties, are R. C. A surface is R. C. iff it is rational. Not all R. C. threefolds (for example: the general quartic hypersurface of \(\mathbb{P}_ 4)\) are expected to be unirational. The difficulty is that all birational invariants (such as \(\pi_ 1\) if \(k=\mathbb{C}\), or \(h^ 0(X,\otimes^ \nu\Omega^ 1_ x))\) which are known to vanish for unirational varieties also vanish for \(X\) which are R.C. By contrast, rational connectedness is deformation invariant. This easily follows from the main result of this paper, which is of fundamental importance for the theory of R. C. varieties: if \(X\) is R. C., then any finite subset of \(X\) is contained in some irreducible rational curve. The argument rests on refined techniques of deformation theory, which also provide the following result (of independent interest, from which the above follows easily): Let \(g:D\to X\) be a nonconstant morphism from a smooth proper curve \(D\) to \(X\), and let \(\overline b_ 1,\ldots,\overline b_ q\) belong to \(g(D)\). Let \(C_ 1,\ldots,C_ N\) be semipositive rational curves on \(X\) (i.e.: \(C_ i=f_ i(\mathbb{P}_ 1)\), with \(f^*_ i(TX)\) semipositive for all \(i\)'s). Assume no \(C_ i\) contains some \(\overline b_ j\). Then: if \(N\) is sufficiently large (depending on \(g^*(TX))\), there exists \(I\subset\{1,\ldots,N\}\) such that the curve \((g(D)+\sum_{i\in I}C_ i)=D_ 0\) moves in an algebraic family of curves \((D_ s)_{s\in S}\), all \(D_ s\) going through \((b_ j)\), and \(D_ s\) being irreducible of the same genus as \(D\) for \(s\) generic in \(S\).
An effective bound for \(N\) above when \(D=\mathbb{P}_ 1\) permitted the authors to show that the family of Fano \(n\)-folds is bounded [cf. Classification of irregular varieties minimal models and abelian varieties, Proc. Conf., Trento/Italy 1990; Lect. Notes Math. 1515, 100- 105 (1992; Zbl 0776.14012)]. The last section contains (among others) the following characterisation of R. C. threefolds: \(X\) is R. C. iff
\[
h^ 0(X,\Omega^ 1_ X)=h^ 0(X,S^ 2(\Omega^ 1_ X))=h^ 0(X,{\mathcal O}_ X(\nu K_ x))=0\quad\text{(for all } \nu>0).
\]
This result needs Mori's minimal model program, as well as Miyaoka's characterization of uniruledness in dimension 3. rational components; Fano \(n\)-folds; connected curve; unirational varieties; rational connectedness; deformation theory; R. C. threefolds J. Kollár, Y. Miyaoka and S. Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429-448. Rational and unirational varieties, Formal methods and deformations in algebraic geometry, Connected and locally connected spaces (general aspects), Special algebraic curves and curves of low genus Rationally connected 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 articles of this volume will be reviewed individually. For the 2012 SMM papers see [Zbl 1290.00026]. Proceedings of conferences of miscellaneous specific interest, Vector bundles on curves and their moduli, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Topological entropy, Structure theory for Lie algebras and superalgebras Papers of the Mexican Mathematical Society | 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 starting point for the present work is the observation that several cohomology theories for algebraic varieties are naturally endowed with a kind of multiplicative structure that goes beyond cup-products. In topology, this is a well-known phenomenon, which is studied under the heading of cohomology operations, called Massey products in the case of interest for us.''
The author obtains higher multiplications on motivic cohomology compatible with the absolute Hodge realization. He obtains DGAs computing absolute Hodge cohomology with real coefficients. A presentation for cohomology classes arising as Massey products is given. Their vanishing in investigated, applications and a state of problems are given. AH cohomology; DGA; minimal model; \(K(\pi,1)\)-conjecture; motivic cohomology; absolute Hodge cohomology Wenger, T.: Massey products in Deligne-cohomology. PhD thesis, Münster University (2000) Secondary and higher cohomology operations in algebraic topology, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Other homology theories in algebraic topology, Motivic cohomology; motivic homotopy theory Massey products in Deligne-cohomology | 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 is a survey on some very important recent works on the adjoint linear system \(K_X +tL\) of a line bundle \(L\) on a smooth complex projective variety \(X\) and on the pluricanonical maps of \(X\) with special stress on the very important joint works by Ein and Lazarsfeld. The reader will find here an algebraic construction of the multiplier ideals and its use to give a simple proof for Reider's theorem (the case \(\dim (X)= 2)\). very ample line bundle; adjoint linear system; multiplier ideals L. Ein, \textit{Adjoint linear systems}, in: \textit{Current Topics in Complex Algebraic Geometry}, MSRI Publications, 1995, pp. 87-95. Divisors, linear systems, invertible sheaves, Surfaces and higher-dimensional varieties Adjoint linear 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 Let \(L\) be a very ample line bundle on a smooth complex projective surface \(S\) with canonical bundle \(K_ S\) and consider the adjoint bundles \({\mathbf L}^{\otimes n}=(K_ S\otimes L)^{\otimes n}\). It is well known from adjunction theory when the maps associated with \({\mathbf L}^{\otimes n}\) are embeddings. The authors prove that for \(n\geq 2\) these embeddings are projectively normal (for \(n=1\) there are counterexamples) and that the ideals of the images of \(S\) are generated by quadrics. The corresponding projective normality result for higher dimensions has been obtained by the first author and \textit{A. J. Sommese} [Comment. Math. Helv. 66, No. 3, 362-367 (1991)] and also proven independently by \textit{A. Bertram}, \textit{L. Ein} and \textit{R. Lazarsfeld} [J. Am. Math. Soc. 4, No. 3, 587-602 (1991)] in connection with the study of syzygies of projective varieties. very ample line bundle; adjoint bundles; projective normality Projective techniques in algebraic geometry, Surfaces and higher-dimensional varieties Projectively normal adjunction 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 \(A\) denote a commutative ring of characteristic \(p>0\). The commutative extensions \(\hbox{Ext}^ 1_ A(\mathbb{G}_{a,A},\mathbb{G}_{m,A})\) and \(\hbox{Ext}^ 1_ A(\hat \mathbb{G}_{a,A},\hat \mathbb{G}_{m,A})\) are explicitly described when \(A\) is a \(\mathbb{Z}_{(p)}\)-algebra. Here \(\mathbb{G}_{a,A}\) (respectively \(\mathbb{G}_{m,A})\) denotes the additive (respectively multiplicative) group over \(A\), and \(W_ A\) the group scheme of Witt vectors over \(A\). The corresponding formal groups are indicated by putting a hat (or roof) on algebraic objects. Let \(W(A)\) denote the ring of Witt vectors over \(A\), and let \(\hat W(A)\) be a subset of \(W(A)\) consisting of Witt vectors \({\mathbf a}=(a_ r)_{r\geq 0}\) such that \(a_ r\) is nilpotent for all \(r\) and \(a_ r=0\) for almost all \(r\). \(F\) stands for the Frobenius endomorphism of \(W(A)\). -- Let
\[
F_ p(U;X,Y)=\exp\left(\sum_{i\geq1}U^{p^ i-1}{X^{p^ i}+Y^{p^ i}-(X+Y)^{p^ i}\over p^ i}\right)\in\mathbb{Z}_{(p)}[U][[X,Y]]
\]
be a formal power series. Associated to \({\mathbf a}=(a_ r)_{r\geq 0}\in W(A)\), define a formal power series
\[
F_ p({\mathbf a};X,Y)=\prod_{r\geq 0}F_ p(a_ r;X,Y)\in A[[X,Y]].
\]
Theorem. Let \(A\) be an \(\mathbb{F}_ p\)-algebra. Then the map \({\mathbf a}\mapsto F_ p({\mathbf a};X,Y)\) induces bijective homomorphisms \(W(A)/F\cong H^ 2_ 0(\hat \mathbb{G}_{a,A},\hat \mathbb{G}_{m,A})\) and \(\hat W(A)/F\cong H^ 2_ 0(\mathbb{G}_{a,A},\mathbb{G}_{m,A})\).
As \(H^ 2_ 0(\hat \mathbb{G}_{a,A},\hat \mathbb{G}_{m,A})\) (respectively \(H^ 2_ 0(\mathbb{G}_{a,A},\mathbb{G}_{a,A}))\) is isomorphic to the group of classes of commutative extensions of \(\hat \mathbb{G}_{a,A}\) by \(\hat \mathbb{G}_{m,A}\) (respectively \(\mathbb{G}_{a,A}\) by \(\mathbb{G}_{m,A})\), which split as extensions of formal \(A\)-schemes (respectively \(A\)-schemes), the above theorem gives an explicit description of \(\hbox{Ext}^ 1_ A(\hat \mathbb{G}_{a,A},\hat \mathbb{G}_{m,A})\) (respectively \(\hbox{Ext}^ 1_ A(\mathbb{G}_{a,A},\mathbb{G}_{m,A}))\). extensions of formal groups; extensions of algebraic groups; Witt vectors; Frobenius endomorphism [SS] Sekiguchi, T., Suwa, N.: A note on extensions of algebraic and formal groups I. Math. Z.206, 567--575 (1991) Formal groups, \(p\)-divisible groups, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Witt vectors and related rings A note on extensions of algebraic and formal groups. I | 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 is about the moduli superstack and superspace of supersymmetric (susy) curves.
Let \(\mathcal{S}\) be the category of complex superspaces and denote by \(\mathcal{S}_{\mathrm{ev}}\) the category of complex spaces (even complex superspaces). Then a 2-category \(\mathrm{FIB}_{\mathcal{S}}\) is defined to be the 2-category of categories fibered in groupoids over \(\mathcal{S}\). Similarly, a 2-category \(\mathrm{FIB}_{\mathcal{S}_{\mathrm{ev}}}\) is a 2-category of categories fibered in groupoids over \(\mathcal{S}_{\mathrm{ev}}\). In this case, we have two following functors.
The first functor is the natural incolusion, \(i:\mathrm{FIB}_{\mathcal{S}_{\mathrm{ev}}}\rightarrow\mathrm{FIB}_{\mathcal{S}}\), defined by
\[
\mathcal{N}\mapsto i(\mathcal{N}):=(\mathcal{C}_{\mathcal{N}}\times\mathcal{S}_{\mathrm{ev}} \, \mathcal{S}, p_2),
\]
where the tensor product of categories is done with respect to the fibration functor \(P_{\mathcal{N}}:\mathcal{C}_{\mathcal{N}}\rightarrow\mathcal{S}_{\mathrm{ev}}\) and the bosonic quotient \(/\Gamma:\mathcal{S}\rightarrow\mathcal{S}_{\mathrm{ev}}\), for canonical automorphism \(\Gamma\), and \(p_2\) is the projection onto the second factor [\textit{G. Codogni} and \textit{F. Viviani}, Adv. Theor. Math. Phys. 23, No. 2, 345--402 (2019; Zbl 07431021)].
The second functor is the bosonic truncation, \((-)_{\mathrm{bos}}:\mathrm{FIB}_{\mathcal{S}}\rightarrow\mathrm{FIB}_{\mathcal{S}_{\mathrm{ev}}}\), defined as follows.
\[
\mathcal{M}\mapsto\mathcal{M}_{\mathrm{bos}}:=(\mathcal{C}_M\times_{\mathcal{S} }\mathcal{S}_{\mathrm{ev}}, p_2),
\]
where the tensor product of categories is done with respect to the fibration functor \(p_{\mathcal{M}}:\mathcal{C}_{\mathcal{M}}\rightarrow \mathcal{S}\) and the natural inclusion \(i\mathcal{S}_{\mathrm{ev}}\rightarrow\mathcal{S}\) [\textit{G. Codogni} and \textit{F. Viviani}, Adv. Theor. Math. Phys. 23, No. 2, 345--402 (2019; Zbl 07431021)].
Let \(\mathrm{sGR}\) be the \(2\)-category of complex supergroupoids, and \(\mathrm{sGR}^\mathrm{et}\) be its full sub \(2\)-category whos objects are etale complex supergroupoids. In a similar way, denote by \(\mathrm{GR}\) the \(2\)-category of complex groupoids and its full sub \(2\)-category of etale complex groupoids by \(\mathrm{GR}^\mathrm{et}\). Then one may have the following.
Proposition [\textit{G. Codogni} and \textit{F. Viviani}, Adv. Theor. Math. Phys. 23, No. 2, 345--402 (2019; Zbl 07431021)]. (i) The natural inclusion \(i:\mathrm{FIB}_{\mathcal{S}_{\mathrm{ev}}}\rightarrow\mathrm{FIB}_{\mathcal{S}}\) sends complex (resp. DM, separated, smooth) stalks into complex (resp. DM, separated, smooth) superstalks.
(ii) The bosonic trunction \((-)_{\mathrm{bos}}:\mathrm{FIB}_{\mathcal{S}}\rightarrow\mathrm{FIB}_{\mathcal{S}_{\mathrm{ev}}}\) sends complex (resp. DM, separated, smooth) superstalks into complex (resp. DM, separated, smooth) stalks.
Proposition [\textit{G. Codogni} and \textit{F. Viviani}, Adv. Theor. Math. Phys. 23, No. 2, 345--402 (2019; Zbl 07431021)]. (i) The association \(i:\mathrm{GR}\rightarrow \mathrm{sGR}\)
\[
\left(Y_1\overset{s}{\underset{t}{\rightrightarrows}}Y_0, u, \iota, m \right)\mapsto \left(i(Y_1)\overset{i(s)}{\underset{i(t)}{\rightrightarrows}}i(Y_0), i(u), i(\iota), i(m) \right),
\]
defines a \(2\)-functor which sends \(\mathrm{GR}^\mathrm{et}\) into \(\mathrm{sGR}^\mathrm{et}\) and such that \(i([Y_1\rightrightarrows Y_0])=[i(Y_1)\rightrightarrows i(Y_0)]\).
(ii) The association \((-)_{\mathrm{bos}}:\mathrm{sGR}\rightarrow\mathrm{GR}\)
\[
\left(X_1\overset{s}{\underset{t}{\rightrightarrows}}X_0, u, \iota, m \right)\mapsto \left((X_1)_{\mathrm{bos}}\overset{s_{\mathrm{bos}}}{\underset{t_{\mathrm{bos}}}{\rightrightarrows}}(X_0)_{\mathrm{bos}}, u_{\mathrm{bos}}, \iota_{\mathrm{bos}}, m_{\mathrm{bos}} \right),
\]
defines a \(2\)-functor which sends \(\mathrm{sGR}^\mathrm{et}\) into \(\mathrm{GR}^\mathrm{et}\) and such that \([X_1\rightrightarrows X_0]_{\mathrm{bos}}=[(X_1)_{\mathrm{bos}} \rightrightarrows (X_0)_{\mathrm{bos}}]\).
(iii) The \(2\)-functor \(i\) is \(2\)-fully faithful and it is left adjoint of the \(2\)-functor \((-)_{\mathrm{bos}}\).
Finally, we have the following theorem.
Theorem [\textit{G. Codogni} and \textit{F. Viviani}, Adv. Theor. Math. Phys. 23, No. 2, 345--402 (2019; Zbl 07431021)]. (i) The association \(-/\Gamma: \mathrm{sGR}^\mathrm{et}\rightarrow \mathrm{GR}^\mathrm{et}\)
\[
\left(X_1\overset{s}{\underset{t}{\rightrightarrows}}Y_0, u, \iota, m \right)\mapsto \left(X_1/\Gamma\overset{s/\Gamma}{\underset{t/\Gamma}{\rightrightarrows}}X_0/\Gamma, u/\Gamma, \iota/\Gamma, m/\Gamma \right),
\]
defines a \(2\)-functor which is left adjoint to the natural incolusion \(i:\mathrm{GR}^\mathrm{et}\rightarrow \mathrm{sGR}^\mathrm{et}\).
(ii) The bosonic trunction preserves weak equivalences, therefore it gives a pseudo-functor \(/\Gamma:\mathrm{sST}^\mathrm{TD}\rightarrow\mathrm{ST}^\mathrm{DM}\),
\[
\mathcal{M}=[X_1\rightrightarrows X_0]\mapsto \mathcal{M}/\Gamma:=[X_1/\Gamma\rightrightarrows X_0/\Gamma],
\]
which is the left adjoint to the natural incolusion \(i:\mathrm{ST}^\mathrm{TD}\rightarrow\mathrm{sST}^\mathrm{DM}\), where \(\mathrm{sST}^\mathrm{TD}\) and \(\mathrm{ST}^\mathrm{DM}\) are \(2\)-categories of Deligne-Mumford (\(\mathrm{DM}\)) complex superstalks and \(\mathrm{DM}\) complex stalks , respectively. Moreover, a \(\mathrm{DM}\) complex superstalk \(\mathcal{M}\) is separated if and only if \(\mathcal{M}/\Gamma\) is separated.
Let \(\mathfrak{M}_g\) be the moduli superstalck os SUSY curves of genus at least 2.
Theorem [\textit{G. Codogni} and \textit{F. Viviani}, Adv. Theor. Math. Phys. 23, No. 2, 345--402 (2019; Zbl 07431021)]. Let \(g\geq 2\)
(1) \(\mathfrak{M}_g\) is a smooth and separated \(\mathrm{DM}\) complex superstalck of dimension \(3g-3|2g-2\) whose bosonic trunction \((\mathfrak{M}_g)_{\mathrm{bos}}\) is the complex stalk \(\mathcal{S}_g\) of spin curves of genus \(g\). Morover, \(\mathfrak{M}_g\) has two connected components, denote by \(\mathfrak{M}_g^+\) and \(\mathfrak{M}_g^-\), whose bosonic trunctions, \((\mathfrak{M}_g)_{\mathrm{bos}}^+\) and \((\mathfrak{M}_g)_{\mathrm{bos}}^-\), are the complex stalks \(\mathcal{S}_g^+\) and \(\mathcal{S}_g^-\) of even and odd spin curves of genus \(g\).
(2) There exists a coarse moduli superspace \(\mathbb{M}_g\) for \(\mathfrak{M}_g\), which is an ordinary complex space and it is also a coarse moduli space for the bosonic quotient \(\mathfrak{M}_g/\Gamma\). The complex space \((\mathbb{M}_g)_{\mathrm{red}}\) is isomorphic to the coarse moduli space \(\mathcal{S}_g\) of spin curves of genus \(g\). In particular, \(\mathbb{M}_g\) is separated and has two connected components whose underlying reduced spaces are the coarse moduli spaces \(\mathcal{S}_g^+\) and \(\mathcal{S}_g^-\) of even and odd spin curves of genus \(g\).
Let \(\pi:\mathcal{C}\rightarrow S\) be a susy curve such that \(\pi_* L=0\). In this case, \(\mathrm{Ber}(\mathcal{C})\) is the Berezinian of the relatibe cotangent bundle [\textit{G. Codogni} and \textit{F. Viviani}, Adv. Theor. Math. Phys. 23, No. 2, 345--402 (2019; Zbl 07431021)]. One may observe that to any global section one can associate a cohomology class in \(H^1(C,\mathbb{C})\). Now if, as a topological space, \(\mathcal{C}\) is homeomorphic to \(C\times S_{\mathrm{bos}}\) for a fibre \(C\), then the integration defines a morphism
\[
\int : \pi_*\mathrm{Ber}(C)\rightarrow H^1(C,\mathbb{Z})\otimes_{\mathbb{Z}}(\mathcal{O}_S)^\Gamma.
\]
It is shown in this paper that the image of this map is an isotropic \(g\)-dimensional subspace of \(H^1(C,\mathbb{Z})\otimes_{\mathbb{Z}}(\mathcal{O}_S)^\Gamma\). This means we have a morphism \(P:S/\Gamma \rightarrow \mathcal{A}_g\) [\textit{G. Codogni} and \textit{F. Viviani}, Adv. Theor. Math. Phys. 23, No. 2, 345--402 (2019; Zbl 07431021)]. This map is functorial, therefore, we have a period map
\[
\tilde{P}:\mathfrak{M}_g^+\dashrightarrow \mathcal{A}_g,
\]
which itself, can be lifted to a morphism
\[
P:\mathfrak{M}_g^+\dashrightarrow \mathcal{N}_g,
\]
where \(\mathcal{N}_g\) is the moduli stack of abelian varieties endowed with a symmetric theta divisor [\textit{G. Codogni} and \textit{F. Viviani}, Adv. Theor. Math. Phys. 23, No. 2, 345--402 (2019; Zbl 07431021)].
Theorem [\textit{G. Codogni} and \textit{F. Viviani}, Adv. Theor. Math. Phys. 23, No. 2, 345--402 (2019; Zbl 07431021)]. Let \(C_L\in \mathfrak{M}_g^+/\Gamma\) be a split susy curve such that \(h^0(L)=0\). 1) The infinitesmal period map at \(C_L\)
\[
d(P/\Gamma)_{C_L}:T_{[C_L]}(\mathfrak{M}_g^+/\Gamma)\rightarrow T_{P(C_L)}\mathcal{N}_g=\mathbb{Sym}^2H^1(C,\mathcal{O}),
\]
where
\[
T_{[C_L]}(\mathfrak{M}_g^+/\Gamma)H^2(C\times C,L^{-1}\boxtimes L^{-1}(-\Delta))^+,
\]
is the even part of the \(H^2\) of the morphism of sheaves on \(C\times C\)
\[
L^{-1}\boxtimes L^{-1}(-\Delta)\hookrightarrow\mathcal{O}
\]
defined by the multiplication with the Szego kernel \(S_L\) associated to \(L\).
2) The infinitesimal period map \(d(P/\Gamma)_{C_L}\) at \(C_L\) is surjective. complex superspace; superstack; Moduli superstack; supersymmetric curves; Kuranishi family Supervarieties, Complex supergeometry, Noncommutative algebraic geometry Moduli and periods of supersymmetric 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 \(X\) denote a compact Riemann surface of genus \(g\geq 2\). A symmetry \(\sigma\) of \(X\) is an antiholomorphic involution of \(X\). If \(X\) admits a symmetry, then we call \(X\) symmetric. The fixed point set of a symmetry \(\sigma\) consists of simple closed curves called the ovals of \(\sigma\). A group \(G\) of conformal automorphisms of \(X\) is called large if \(X/G\) has genus zero and the map \(\pi : X \rightarrow X/G\) is branched over precisely three points. In the paper under review, the author provides formulas for the number of ovals of a symmetry \(\sigma\) of a compact Riemann surface \(X\) which admits a large group of automorphisms \(G\).
The author proves the results using non-Euclidean crystallographic (NEC) groups and uniformization. Suppose that \(G\) is a large group of automorphisms of \(X\). In \textit{D. Singerman}, [Math. Ann. 210, 17--32 (1974; Zbl 0272.30022)] necessary and sufficient algebraic conditions on a generating set for \(G\) were provided for \(X\) to be symmetric. This result together with a formula for the number of ovals of a symmetry from \textit{G. Gromadzki} [J. Pure Appl. Algebra 121, No.3, 253--269 (1997; Zbl 0885.14026)] which is adapted to the special case where \(G\) is a large automorphism group, allows the author to explicitly calculate the number of ovals of a symmetry of \(X\). The paper is well written. Symmetries of surfaces Gromadzki G.: On Singerman symmetries of a class of Belyi Riemann surfaces. J. Pure Appl. Algebra 213, 1905--1910 (2009) Compact Riemann surfaces and uniformization, Riemann surfaces; Weierstrass points; gap sequences On Singerman symmetries of a class of Belyi Riemann 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 \(k\) be any field, \(\text{Hilb}^{p(z)}_{\mathbb{P}^n_k}\) the Hilbert scheme of subschemes of \(\mathbb{P}^n\) with Hilbert polynomial \(p(z)\). \textit{F. J. Macaulay} showed [Proc. Lond. Math. Soc. (2) 26, 531-555 (1927; JFM 53.0104.01)] that there exists a unique saturated lexicographic ideal \(L\) such that \(k[x_0, \dots,x_n]/L\) has Hilbert polynomial \(p(z)\). We show that the scheme corresponding to \(L\) is parametrized by a smooth point on the Hilbert scheme. In the process we calculate the dimension of the unique component through this point explicitly, and describe explicitly the subscheme corresponding to the general point of this component. smoothness; lexicographic point; Hilbert scheme; JFM 53.0104.01; Hilbert polynomial A. Reeves - M. Stillman, Smoothness of the lexicographic point. J. Algebraic Geom., 6 (2) (1997), pp. 235-246. Zbl0924.14004 MR1489114 Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Smoothness of the lexicographic point | 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 \(\mathcal M_{g,n}\) be the moduli space of smooth genus \(g\) curves with \(n\) ordered marked points. The symmetric group \(S_n\) acts naturally on \(\mathcal M_{g,n}\) by permuting the marked points. The main result of the paper is a formula of the generating function of the \(S_n\) equivariant Euler characteristics of \(\mathcal M_{g,n}\). The author first derives a formula of the equivariant Euler characteristics of a configuration space with a finite group action, and then uses it to prove the main result. The coefficients involved in the main formula correspond to the orbifold Euler characteristics of moduli spaces of curves with a specific type of automorphism. These Euler characteristics are computed using a result of \textit{J. Harer} and \textit{D. Zagier} [Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)]. moduli spaces; equivariant Euler characteristic; orbifold Euler characteristic DOI: 10.1016/j.aim.2013.10.003 Families, moduli of curves (algebraic), Fine and coarse moduli spaces The equivariant Euler characteristic of moduli spaces 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 Auszug aus einem Vortrage, den Herr Ahlborn im Anschlusse an die Arbeiten von Clebsch (Crelle J. LXIII. und Theorie der Abel'schen Functionen von Clebsch und Gordan) gehalten hat. Analytic theory of abelian varieties; abelian integrals and differentials On the significance of the number \(p\) in the abelian functions and their connections with geometry. | 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 Nachdem der Verfasser die Arbeiten von Cayley und Salmon über höhere Berührung erwähnt hat, giebt er an, auf welche Weise er die Aufgabe behandelte, den Kegelschnitt zu finden, der in einem Punkte \(M_0\) einer gegebenen Curve eine Berührung von bestimmter Ordnung eingeht. Er betrachtet zuerst einen Kegelschnitt, der in \(M_0\) den Polarkegelschnitt dieses Punktes berührt, und bildet die Gleichung der Geraden, die durch \(M_0\) und die Schnittpunkte des ersten Kegelschnitts mit der Curve gehen. Diese Gleichung wird explicit aufgestellt und dann die Bedingungen, damit der Kegelschnitt eine Berührung irgend einer Ordnung mit der Curve habe, wobei letztere auf drei Gerade bezogen ist, von denen die eine die Tangente in \(M_0\) ist; die andere ist die gemeinschaftliche Sehne der beiden genannten Kegelschnitte, und die dritte ist eine belibige durch \(M_0\) gehende Gerade. Weiterhin findet sich noch eine Anwendung auf Curven dritten Grades.
In der zweiten Arbeit bezieht sich der Verfasser auf eine Berührung fünfter Ordnung, und gelangt zu dem Resultat, dass dann der Punkt \(M_0\) auf einer Curve vom Grade \(12m-27\) liegen muss, wenn \(m\) der Grad der vorliegenden Curve ist.
In einer dritten Arbeit wird die Gleichung des Kegelschnitts, der die Berührung fünfter Ordnung mit der gegebenen Curve eingeht, aufgestellt und dies auf eine Curve vierten Grades angewandt. conics; curve theory Questions of classical algebraic geometry, Plane and space curves Search for the conditions under which a conic and a given curve have a contact of specific order | 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 Herr R. geht von den Jacobi'schen Sätzen über die Durchdringungscurven und Schnittpunkte von 2 oder 3 algebraischen Flächen (Crelle XV 299) aus, begründet dieselben zum Theil rein geometrisch und entwickelt dann eine grosse Anzahl neuer Sätze welche über die projectivische Erzeugung algebraischer Raumcurven und algebraischer Flächen wichtige Aufschlüsse geben. Bedeutet \(F_p\) eine algebraische Fläche \(p^{\text{ter}}\) Ordnung, \(C^{pq}\) die Durchschnittscurve einer \(F^p\) und \(F^q\) \([n,p,q]\) die Gruppe der \(n.p.q\) Schnittpunkte von \(F_p\), \(F^q\), \(F^n\), so stellt Herr R. zunächst die Anzahl von Punkten fest, welche zur Bestimmung einer \(C^{pq}\) auf einer \(F_p\) gegeben sein müssen; daraus folgt die Zahl der Punkte, die zur Bestimmung eines auf \(F_p\) gelegenen Curvenbüschels \(np^{\text{ter}}\) Ordnung erforderlich sind, resp. zur Bestimmung einer Punktgruppe \([n\,n\,p]\). Zwei auf derselben \(F_p\) gelegene Curven \(C^{pq}\) und \(C^{pn}\) schneiden sich in der Punktgruppe \([p,q,n]\). -- Die Betrachtung der Flächen und Curven, welche durch eine solche gelegt werden können, füllt ein ergebnissreiches Capitel, auf das sich folgende Sätze über die projectivische Erzeugung der algebraischen Raumcurven und Flächen gründen: ``Die allgemeine auf einer Fläche \(q^{\text{ter}}\) Ordnung gelegene Curve \((n+p)q^{\text{ter}}\) Ordnung kann nicht durch projectivische Curvenbüschel \(p^{\text{ter}}\) und \(n^{\text{ter}}\) Ordnung erzeugt werden, sondern nur die besondere Art derselben, welche eine Punktgruppe \([n,n,q]\) enthält; aber für \(s>2\) entsteht jede \(C^{2.s}\) auf unzählig viele Arten durch zwei projectivische Curvenbüschel \(2.2^{\text{ter}}\) und \(2(s-2)^{\text{ter}}\) Ordnung''. ``Enthält eine Fläche \(n+p^{\text{ter}}\) Ordnung eine Gruppe \([n,n,n]\) (ob die allgemeine \(F^{n+p}\) eine solche immer enthalten muss, hat Herr R. noch nicht ermitteln können), so lässt sie sich durch zwei reciproke Flächenbündel \(n^{\text{ter}}\) und \(p^{\text{ter}}\) Ordnung erzeugen, jede \(F^s\) entsteht durch Bündel \(1^{\text{ter}}\) und \(s-1^{\text{ter}}\), \(2^{\text{ter}}\) und \(s- 2^{\text{ter}}\) Ordnung.''
In Betreff der Ergebnisse, welche die Theorie der Polaren aus den angedeuteten Sätzen liefert, möge auf die wichtige und inhaltreiche Abhandlung selbst verwiesen werden. algebraic surfaces; algebraic curves Questions of classical algebraic geometry, Surfaces and higher-dimensional varieties On algebraic surfaces, their intersection curves, intersection points, and projective methods of generation | 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 \({\mathcal M}_ 2\) be the moduli space of curves of genus 2 and \({\mathcal S}\) be the moduli space of ''nodal cubics'' i.e. of cubic surfaces with one ordinary double point. We produce a birational map \({\mathcal S}\to {\mathcal M}_ 2\) as follows: to a nodal cubic S with note P we associate a genus 2 curve \(\Gamma_ S\) which is the double cover of the conic \({\mathbb{P}}roj(\tan gent\quad cone\quad to\quad S\quad at\quad P)\) branched at the six points which are the lines through P lying on S; conversely, the bicanonical image of a genus 2 curve \(\Gamma\) is a conic in \({\mathbb{P}}^ 2\) with six distinguished points on it and this determines a nodal cubic. By using the classical Sylvester's theorem which gives a unique ''canonical'' equation for a generic (even nodal) cubic surface we construct an explicit ''birational model'' W for \({\mathcal S}\). W is a hypersurface of \({\mathbb{P}}^ 4/\sigma_ 5\) where \(\sigma_ 5\) is the symmetric group on 5 letters. The rationality of W is then proved by some computations, thus yielding the rationality of \({\mathcal S}\) and \({\mathcal M}_ 2\) over an algebraically closed field of characteristic 0. rationality of moduli space of curves of genus 2; rationality of moduli space of nodal cubics Rational and unirational varieties, Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory, Algebraic moduli problems, moduli of vector bundles Nodal cubic surfaces and the rationality of the moduli space of curves of genus two | 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 0563.00007.]
Let \(K=F(x_ i: i\in \Gamma)\) be a finite rank transcendental extension of a field \(F\) and let \(G\) be a finite group acting transitively on \(\{x_ i: i\in \Gamma \}\), so \(G\) acts on \(K\). Let \(F(\Gamma,G)\) denote the invariant subfield of \(K\). Noether's problem is to determine for which \(F\), \(\Gamma\) and \(G\) \(F(\Gamma,G)/F\) is rational, that is, purely transcendental.
This paper is a succinct survey of the results known to date, and their relationship to other problems in Galois theory. The most important results are a complete answer for \(G\) abelian and a class of examples, due to the author, of non-rational extensions in the case that \(F\) is algebraically closed. group invariants; retract rational; Galois extensions; approximation property; finite rank; transcendental extension; Noether's problem; survey David J. Saltman,Groups Acting on Fields: Noether's Problem, Contemp. Mathematics43 (1985), 267--277 Separable extensions, Galois theory, Rational points, Representations of groups as automorphism groups of algebraic systems Groups acting on fields: Noether's problem | 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 JFM 07.0450.03 parabola; fourth order curves; nodes; doublepoints Plane and space curves Solution de la question 22. | 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 calculates the integral intersection homology groups of weighted projective spaces and pseudo-lens spaces. Most computations of intersection homology have been done for the rational groups. The groups calculated here have interesting torsion. integral intersection homology; weighted projective spaces; pseudo-lens spaces; torsion Algebraic topology on manifolds and differential topology, Products and intersections in homology and cohomology, Classical real and complex (co)homology in algebraic geometry, \(n\)-folds (\(n>4\)) Intersection homology of weighted projective spaces and pseudo-lens 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 Let \(k\) denote an algebraically closed field. Let \(Y\) be a nonsingular projective curve over \(k\) and let \(f:X\rightarrow Y\) be a semistable curve of genus \(g\geq 2\) such that \(X\) is smooth and the generic fiber of \(f\) is a smooth hyperelliptic curve. When \(k=\mathbb C\), \textit{M.~Cornalba} and \textit{J.~Harris} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, No.3, 455--475 (1988; Zbl 0674.14006)] established a formula for the class of the \((8g+4)\)th tensor power of the Hodge bundle of this family in terms of the classes of the singular fibers. This formula was extended to the case when the characteristic of \(k\) is greater than two by \textit{I.~Kausz} [Compos. Math. 115, No.1, 37--69 (1999; Zbl 0934.14015)]. The author shows that the Cornalba-Harris formula also holds when the characteristic of \(k\) is two. The strategy of the proof is to use the result in characteristic zero and a compactification \(\overline{\mathcal I}_g\) of the algebraic stack (over \(\mathbb Z\)) of smooth hyperelliptic curves of genus \(g\) to derive the result in all characteristics. A key element in the proof is a result of \textit{S. Maugeais} [Espace de modules des courbes hyperelliptiques stables et une inégalité de Cornalba-Harris-Xiao, preprint, \texttt{http://arxiv.org/abs/math.AG/0107015}] that is used to establish the irreducibility of the specialization of \(\overline{\mathcal I}_g\) to characteristic two. algebraic stack; Hodge class Yamaki K.: Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic. Asian J. Math. 8(3), 409--426 (2004) Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Picard groups Cornalba-Harris equality for semistable hyperelliptic curves in positive 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 For G a finite group of polynomial automorphisms of \({\mathbb{C}}^ n\) the author shows that there is a nonsingular algebraic compactification \(\iota\) : \({\mathbb{C}}^ n\to X\) with \(A=X-\iota ({\mathbb{C}}^ n)\) such that \(G=\iota^{-1}\circ \hat G\circ \iota\) for a finite subgroup \(\hat G\) of birational and biregular automorphisms of X which stabilize A. The leads to another proof [see e.g. the author, Mem. Fac. Sci., Kyushu Univ., Ser. A 36, 85-105 (1982; Zbl 0547.32015) and 37, 45-56 (1983; Zbl 0547.32016)], of the fact that for \(n=2\) such a G is conjugate to a subgroup of GL(2,\({\mathbb{C}})\). The author also shows that for \(n=3\), if X is the quadric, then G is conjugate to a subgroup of GL(3,\({\mathbb{C}})\). characterization of finite group of polynomial automorphisms; algebraic compactification Furushima, M.: Finite groups of polynomial automorphisms in ? n . Tohoku Math. J.35, 415-424 (1983) Complex Lie groups, group actions on complex spaces, Group actions on varieties or schemes (quotients) Finite groups of polynomials automorphisms in \({\mathbb{C}}^ n\) | 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 This part is a survey (nice, but with few proofs and written in 1984) about the complements of an arrangement of hyperplanes (most of the time for Coxeter arrangements); since everything is defined, it is useful also for beginners. complements of an arrangement of hyperplanes; Coxeter arrangements Projective and enumerative algebraic geometry, Reflection groups, reflection geometries, Foundations of classical theories (including reverse mathematics) Arrangements d'hyperplans. I: Les groupes de réflexions. (Arrangement of hyperplanes. I: The reflection groups) | 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 article contains a survey of results on ``generic singularities''. Many of the results discussed were obtained by the author. No proofs are presented.
More precisely, generic singularities are those that appear when we consider a sufficiently general linear projection \(\pi\) of a smooth \(r\)-dimensional projective variety \(X \subset {\mathbb P}^n\) into a linear subspace \( V \) (\( \approx {\mathbb P}^{m}\)) of \( {\mathbb P}^n\), where \( r+1 \leq m \leq 2r\), and (letting \(Y=\pi (X)\)) we exclude points of \(Y=\pi (X)\) in a suitable lower dimensional subvariety. Expanding results of M. Noether, E. Lluis, etc, J. Roberts developed the basic theory of these singularities in the 1970's, see \textit{J. Roberts} [Trans. Am. Math. Soc. 212, 229--268 (1975; Zbl 0314.14003)]. After briefly reviewing this theory, Zaare-Nahandi discusses some of his contributions. For instance, using the notation above, if \(y=\pi(x), x \in X\), is an analytically irreducible generic singularity, he has obtained a very explicit description of the induced homomorphism \(\pi ^* : {\hat {\mathcal O}}_{V,y} \to {\hat {\mathcal O}}_{X,x}\) and of the \textit{local defining ideal} of the singularity \(y\), namely Ker(\(\pi ^*\)). The local defining ideal is expressed in terms of minors of an associated matrix \({\mathcal M}\) with coefficients in \({\hat {\mathcal O}}_{V,y}\). The ring \({\hat {\mathcal O}}_{V,y}\) is isomorphic to a power series ring, and specializing some of the variables the matrix \(\mathcal M\) induces a matrix \({\mathcal M}_0\) with interesting properties. For instance, the defining ideal becomes a square-free monomial ideal. A simplicial complex may be associated to it, some of its properties are studied. As an application, a formula for the depth of \({\mathcal O}_{Y,y}\) is obtained and a partial answer to a conjecture of Andreotti, Bombieri and Holme weak normality of certain points of \(Y\) is gotten.
The author works over an algebraically closed field but if the characterisitc is positive the embedding \(X \subset {\mathbb P}^n\) must satisfy some extra conditions. generic projections; generic singularities; local defining ideal Singularities in algebraic geometry, Local theory in algebraic geometry, Projective techniques in algebraic geometry, Determinantal varieties, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes Algebraic properties of generic singularities | 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 Following Sen, we study the counting of (`twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with \( \mathcal{N} = 4 \) supersymmetry. The generating functions of half-BPS states, twisted as well as untwisted, are given in terms of multiplicative eta products with the Mathieu group, \(M_{24}\), playing an important role. These multiplicative eta products enable us to construct Siegel modular forms that count twisted quarter-BPS states.{
}The square-roots of these Siegel modular forms turn out be precisely a special class of Siegel modular forms, \textit{the dd-modular forms}, that have been classified by Clery and Gritsenko. We show that each one of these dd-modular forms arise as the Weyl-Kac-Borcherds denominator formula of a rank-three Borcherds-Kac-Moody Lie superalgebra. The walls of the Weyl chamber are in one-to-one correspondence with the walls of marginal stability in the corresponding CHL model for twisted dyons as well as untwisted ones. This leads to \textit{a periodic table} of BKM Lie superalgebras with properties that are consistent with physical expectations. black holes in string theory; D-branes Govindarajan S 2011 BKM Lie superalgebras from counting twisted CHL dyons \textit{J. High Energy Phys.}JHEP05(2011) 089 String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Supersymmetric field theories in quantum mechanics, Black holes, Spinor and twistor methods applied to problems in quantum theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Moduli, classification: analytic theory; relations with modular forms, Superalgebras BKM Lie superalgebras from counting twisted CHL dyons | 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. Grothendieck constructed the Hilbert scheme \(\text{Hilb}^n_X\) of \(n\) points on \(X\), for any quasi-projective scheme \(X\) on a noetherian base scheme \(S\). In the paper under review, the authors are interested in showing the existence of the Hilbert scheme of \(n\) points on \(\text{Spec}({\mathcal O}_{X,P})\), where \(P\) is a (non necessarily closed) point on such a scheme \(X\). A natural candidate would be \(\bigcap_{P\in U_\alpha}\text{Hilb}^n_{U_\alpha}\), where \(U_\alpha\) varies in the set of open subsets of \(X\) containing \(P\). But in general an infinite intersection of open subschemes of a scheme is not a scheme. It is a scheme if one takes only locally principal open subschemes.
The authors introduce and study the notion of generalized fraction rings and localized subschemes, of which \(\text{Spec}({\mathcal O}_{X,P})\) is a particular case. They prove that, if \(X\) is a scheme such that \(\text{Hilb}^n_X\) exists, then the functor of points of a localized scheme \({\mathcal S}^{-1}X\) is representable. As a particular case, they get the following result:
If \(X\rightarrow S\) is a projective morphism of Noetherian schemes and \(P\) is a point in \(X\), then the Hilbert scheme of \(n\) points on \(\text{Spec}({\mathcal O}_{X,P})\) exists and coincides with the intersection of the Hilbert schemes of \(n\) points of the open subschemes of \(X\) containing \(P\). localized schemes; determinants; fraction rings Parametrization (Chow and Hilbert schemes) Infinite intersections of open subschemes and 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 This article can be seen as a sequel to the first author's article [J. Algebr. Geom. 14, No. 4, 761-787 (2005; Zbl 1120.14002)], where he calculates the total Chern class of the Hilbert schemes of points on the affine plane by proving a result on the existence of certain universal formulas expressing characteristic classes on the Hilbert schemes in term of Nakajima's creation operators. The purpose of this work is (at least) two-fold. First of all, we clarify the notion of ``universality'' of certain formulas about the cohomology of the Hilbert schemes by defining a universal algebra of creation operators. This helps us to reformulate and extend a lot of the first author's previous results in a very precise manner.
Secondly, we are able to extend the previously found results by showing how to calculate any characteristic class of the Hilbert scheme of points on the affine plane in terms of the creation operators. In particular, we have included the calculation of the total Segre class and the square root of the Todd class. Using these methods, we have also found a way to calculate any characteristic class of any tautological sheaf on the Hilbert scheme of points on the affine plane. This in fact gives another complete description of the ring structure of the cohomology spaces of the Hilbert schemes of points on the affine plane. Samuel Boissière and Marc A. Nieper-Wißkirchen, Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces, J. Algebra 315 (2007), no. 2, 924 -- 953. Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Cycles and subschemes Universal formulas for characteristic classes on 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 This paper derives some orbifold Hurwitz-Hodge integral identities from the Laplace transform of the cut-and-join equation for the orbifold Hurwitz numbers. Hurwitz-Hodge integrals; orbifold Hurwitz numbers; cut-and-join equation; Laplace transformation Families, moduli of curves (analytic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumeration in graph theory Hurwitz-Hodge integral identities from the cut-and-join equation | 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 \(S\) is a surface of general type, then by a result of \textit{A. Beauville} [Invent. Math. 55, 121--140 (1979; Zbl 0403.14006)], if the canonical map \(\phi _S\) is generically finite, then its degree is at most \(36\), if moreover \(p_g(S)\geq 30\), then the degree of \(\phi _S\) is at most \(9\). If \(X\) is a complex projective Gorenstein \(3\)-fold of general type with locally factorial terminal singularities such that the canonical map \(\phi _X\) is generically finite then (by a result of the reviewer [Proc. Japan Acad., Ser. A 80, No. 8, 166--167 (2004; Zbl 1068.14046)]) the degree of \(\phi _X\) is at most \(576\). In the paper under review, the author proves that if moreover \(p_g(X)>105411\), then the degree of \(\phi _X\) is at most \(72\). J.-X. Cai, Degree of the canonical map of a Georenstein \(3\)-fold of general type, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1565--1574. \(3\)-folds, Rational and birational maps, \(4\)-folds Degree of the canonical map of a Gorenstein 3-fold of general 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 This rather important paper indicates a precise concrete way to perform computations in the quantum equivariant ``deformation'' of the cohomology ring of \(G(k,n)\), the complex Grassmannian variety parametrizing \(k\)-dimensional vector subspaces of \({\mathbb C}^n\). It relies on the results of another important paper, regarding the same subject, by the same author [Adv. Math. 203, 1--33 (2006; Zbl 1100.14045)]. The usual singular cohomology ring of \(G(k,n)\) is a very well known object, studied since Schubert's time, at the end of the XIX Century. First of all, it is a finite free \({\mathbb Z}\)-module generated by the so-called Schubert cycles. Furthermore, the special Schubert cycles, the Chern classes of the universal quotient bundle over \(G(k,n)\), generates it as a \({\mathbb Z}\)-algebra. Multiplying two Schubert cycles then amounts to know how to multiply a special Schubert cycle with a general one (Pieri's formula) and a way to express any Schubert cycle as an explicit polynomial expression in the special Schubert cycles (Giambelli's formula).
The obvious way to deform the cohomology of a Grassmannian is to consider the cohomology of the total space of a Grassmann bundle, parametrizing \(k\)-planes in the fibers of a rank \(n\) vector bundle, which is a deformation of the cohomology of any fiber of it. In the last few decades, however, other ways to deform the cohomology ring of \(G(k,n)\) have been studied. \textit{E. Witten} [in: Geometry, topology and physics for Raoul Bott. Lectures of a conference in honor of Raoul Bott's 70th birthday, Harvard University, Cambridge, MA, USA 1993. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Geom. Topol. 4, 357--422 (1995; Zbl 0863.53054)], introduced the small quantum deformation of the cohomology ring of the Grassmannian, whose structure constants were first determined by \textit{A. Bertram} [Adv. Math. 128, No. 2, 289--305 (1997; Zbl 0945.14031)]. Finally, \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)], studied the equivariant deformation of the cohomology of the Grassmannians via the combinatorics of puzzles.
In the beautiful paper under review the author recovers the quantum and equivariant Schubert calculus within a unified framework. Basing on the algebraic properties of the Schur factorial functions, the author realizes the equivariant quantum cohomology ring in terms of generators and relations and gives an explicit basis of polynomial representatives for the equivariant quantum Schubert classes. An alternative approach is offered by \textit{D. Laksov} [Adv. Math. 217, 1869--1888 (2008; Zbl 1136.14042)], where the author proves that the basic results of equivariant Schubert calculus, the basis theorem, Pieri's formula and Giambelli's formula can be obtained from the corresponding results of a more general and elementary framework, as in [\textit{D. Laksov}, Indiana Univ. Math. J., 56, No. 2, 825--845 (2007; Zbl 1136.14042)], by a change of basis.
The paper is organized as follows. Section 1 is the introduction, where the main results are clearly stated and motivated; Section 2 is a useful and very pleasant review of the algebra of factorial Schur functions. The quantum equivariant cohomology of Grassmannians is treated in Section 3, while the proof of the theorem about the presentation of the quantum equivariant cohomology ring is given in Section 4. Section 5 ends the paper with the discussion and the proof of Giambelli's formula in equivariant quantum cohomology. Giambelli's formulas; quantum equivariant Schubert calculus; factorial Schur functions L.C. Mihalcea, \textit{Giambelli formulae for the equivariant quantum cohomology of the Grassmannian}, \textit{Trans. AMS}\textbf{360} (2008) 2285 [math/0506335]. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Giambelli formulae for the equivariant quantum cohomology of the 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 find upper and lower bounds of the multiplicities of irreducible admissible representations \({\pi}\) of a semisimple Lie group \(G\) occurring in the induced representations \(\mathrm{Ind}_H^G{\tau}\) from irreducible representations \({\tau}\) of a closed subgroup \( H\). As corollaries, we establish geometric criteria for finiteness of the dimension of \(\mathrm{Hom}_G({\pi}, \mathrm{Ind}_H^G{\tau})\) (induction) and of \(\mathrm{Hom}_H({\pi}|_H,{\tau})\) (restriction) by means of the real flag variety \(G/P\), and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety. real reductive group; admissible representation; multiplicity; hyperfunction; unitary representation; spherical variety; symmetric space Kashiwara, M.: On the maximally overdetermined system of linear differential equations. I. Publ. Res. Inst. Math. Sci. \textbf{10}, 563-579 (1974/75) Semisimple Lie groups and their representations, Compactifications; symmetric and spherical varieties Finite multiplicity theorems for induction and restriction | 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. Singularities; Geometry; Topology; Sapporo (Japan); Proceedings Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to manifolds and cell complexes, Proceedings, conferences, collections, etc. pertaining to global analysis Singularities--Sapporo 1998. Proceedings of the international symposium on singularities in geometry and topology, Sapporo, Japan, July 6--10, 1998 | 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 Koszul algebra of finite global dimension we define its higher zigzag algebra as a twisted trivial extension of the Koszul dual. If our original algebra is the path algebra of a tree-type quiver, this construction recovers the zigzag algebras of Huerfano-Khovanov. We study examples of higher zigzag algebras coming from Iyama's type A higher representation finite algebras, give their presentations by quivers and relations, and describe relations between spherical twists acting on their derived categories. We connect this to the McKay correspondence in higher dimensions: if \(G\) is a finite abelian subgroup of \(\mathrm{SL}_{d+1}\) then these relations occur between spherical twists for \(G\)-equivariant sheaves on affine \((d+1)\)-space. trivial extension; braid group action; spherical twist; quiver; derived category; Koszul algebra; cluster tilting; equivariant sheaves Quadratic and Koszul algebras, Representations of quivers and partially ordered sets, Derived categories, triangulated categories, Derived categories and associative algebras, ``Super'' (or ``skew'') structure, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry Higher zigzag algebras | 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 extend and generalize results of \textit{C. Scheiderer} [Manuscr. Math. 119, No. 4, 395--410 (2006; Zbl 1120.14047)] on the representation of polynomials nonnegative on two-dimensional basic closed semialgebraic sets. Our extension covers some situations where the defining polynomials do not satisfy the transversality condition. Such situations arise naturally when one considers semialgebraic sets invariant under finite group actions. J. Cimprič, S. Kuhlmann, M. Marshall: Positivity in Power Series Rings, Advances in Geometry, to appear. Formal power series rings, Semialgebraic sets and related spaces, Group actions on varieties or schemes (quotients), Linear algebraic groups over the reals, the complexes, the quaternions Positivity in power series 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 Hurwitz spaces are spaces of pairs \((S,f)\) where \(S\) is a Riemann surface and \(f:S\to\widehat{\mathbb C}\) a meromorphic function. In this work, we study one-dimensional Hurwitz spaces \({\mathcal H}^{D_p}\) of meromorphic \(p\)-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of \((p-1)/2\) transpositions and the monodromy group
is the dihedral group \(D_p\). We prove that the completion \(\overline{{\mathcal H}^{D_p}}\) of the Hurwitz space \({\mathcal H}^{D_p}\) is uniformized by a non-normal index \(p+1\) subgroup of a triangular group with signature \((0;[p,p,p])\). We also establish the relation of the meromorphic covers with elliptic functions and show that \({\mathcal H}^{D_p}\) is a quotient of the upper half plane by the modular group \(\Gamma(2)\cap\Gamma_0(p)\). Finally, we study the real forms of the Belyi projection \(\overline{{\mathcal H}^{D_p}}\to\widehat{\mathbb C}\) and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated. Costa, AF; Izquierdo, M.; Riera, G., One-dimensional Hurwitz spaces, modular curves, and real forms of Belyi meromorphic functions, Int. J. Math. Math. Sci., 2008, 1-18, (2008) Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) One-dimensional Hurwitz spaces, modular curves, and real forms of Belyi meromorphic functions | 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 functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset. equivariant intersection cohomology; ring structure; hypertoric variety Braden, T.; Proudfoot, N., The hypertoric intersection cohomology ring, Invent. Math., 177, 2, 337-379, (2009) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Toric varieties, Newton polyhedra, Okounkov bodies The hypertoric intersection cohomology ring | 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 the authors prove the following result: for any positive integer \(n\) there exists an integer \(r_n\) such that if \(X\) is a smooth projective variety of general type and dimension \(n\), then the \(r\)-canonical map of \(X\) is birational onto its image for all \(r\geq r_n\). pluricanonical map; variety of general type; birational map Christopher D. Hacon & James McKernan, ``Boundedness of pluricanonical maps of varieties of general type'', Invent. Math.166 (2006) no. 1, p. 1-25 Minimal model program (Mori theory, extremal rays), Rational and birational maps Boundedness of pluricanonical maps of varieties of general 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 In the present paper the authors prove the homological mirror symmetry conjecture for Del Pezzo surfaces. Recall that a Landau-Ginzburg model is a pair \((M,W)\), where \(M\) is a non-compact (symplectic and/or complex) manifold and \(W\) is a complex-valued function on \(M\). The idea is that when such a model \((M,W)\) is mirror to a Fano variety \(X\), the complex geometry on \(X\) corresponds to the symplectic geometry of \(M\) and vice versa. The homological mirror symmetry conjecture states that the bounded derived category of coherent sheaves on \(X\) is equivalent to the derived categories of Lagrangian vanishing cycles of \(W\).
Let \(X_K\) be a Del Pezzo surface obtained blowing up \(\mathbb{P}^2\) at \(k\) points. The conjecture is proved by taking as mirror an elliptic fibration \(W_k : M_k \to \mathbb{C}\) with \(k+3\) singular fibers and suitable symplectic form \([B + i \omega]\). Moreover, given a general noncommutative deformation of \(X_K\), there exists a complexified symplectic form \([B + i \omega]\), for which the deformed derived category of \(X_K\) is equivalent to the derived category of Lagrangian vanishing cycles. Conversely, for a generic choice of \([B + i \omega]\), the derived category of Lagrangian vanishing cycles is equivalent to the derived category of coherent sheaves of a noncommutative deformation of a Del Pezzo surface.
In order to prove that, the authors describe in a first time the derived category of \(X_K\) by detailing a strong exceptional collection of objects and morphisms between them. Derived categories of simple degenerations and noncommutative deformations can be described by working on such sets.
In a second time, the construction of the mirror Landau-Ginzburg model is given. The mirror of a given \(X_K\) is an elliptic fibration \(W_k : M_k \to \mathbb{C}\) with \(k+3\) nodal fibers. Moreover, it compactifies to an elliptic fibration \(\bar{W}_k\) over \(\mathbb{P}^1\), in which the fiber above infinity consists of \(9-k\) rational components, and which can be obtained as a deformation of the elliptic fibration \(\bar{W}_0 : \bar{M} \to \mathbb{P}^1\) compactifying the mirror of \(\mathbb{P}^2\). The manifold \(M_k\) is equipped with a symplectic form \(\omega\) and a B-field \(B\) whose cohomology classes are explicitely given by the set of points \(K\).
Once detailed such a construction, the authors recall the definition given in [\textit{P. Seidel}, Proc. 3rd European Congr. Math., Barcelona 2000, II. Progr. Math., 202, 65--85 (2001; Zbl 1042.53060)] of the category of Lagrangian vanishing cycles of a Landau-Ginzburg model, describe explicitely the derived category for \(\bar{W}_k\) and the cohomology class \([B+i\omega]\). This easily leads to prove the required equivalence of categories. homological mirror symmetry; derived category; Del Pezzo surfaces; Landau-Ginzburg model; Lagrangian vanishing cycles; noncommutative deformations D. Auroux, L. Katzarkov, and D. Orlov, ''Mirror Symmetry for Del Pezzo Surfaces: Vanishing Cycles and Coherent Sheaves,'' Invent. Math. 166(3), 537--582 (2006); arXiv:math/0506166. Calabi-Yau manifolds (algebro-geometric aspects), Rational and ruled surfaces, Fano varieties, Families, fibrations in algebraic geometry Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves | 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 main objects of uniformization of the curve \(y^2=x^5-x\) are studied: its Burnside's parametrization, corresponding Schwarz's equation, and accessory parameters. As a result we obtain the first examples of solvable Fuchsian equations on torus and exhibit number-theoretic integer \(q\)-series for uniformizing functions, relevant modular forms, and analytic series for holomorphic abelian integrals. A conjecture of Whittaker for hyperelliptic curves and its hypergeometric reducibility are discussed. We also consider the conversion between Burnside's and Whittaker's uniformizations. {
\copyright 2009 American Institute of Physics} solvable Fuchsian equations; integer \(q\)-series for uniformizing functions; analytic series for holomorphic abelian integrals Brezhnev, Yu.V.: On uniformization of Burnside's curve y2=x5 - x. J. math. Phys. 50, No. 10 (2009) Compact Riemann surfaces and uniformization, Elliptic curves over global fields, Special algebraic curves and curves of low genus, Basic hypergeometric integrals and functions defined by them On uniformization of Burnside's curve \(y^2 = x^5 - x\) | 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. Surfaces; Classification; Meeting; Proceedings; Cortona (Italy) Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to algebraic geometry Problems in the theory of surfaces and their classification. Papers from the meeting held at the Scuola Normale Superiore, Cortona, Italy, October 10-15, 1988 | 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 develop a dynamical scaling limit from rational dynamics to automata in tropical geometry. We compare these dynamics and induce uniform estimates of their orbits. We apply these estimates to introduce a comparison analysis of theory of automata groups in geometric group theory with analysis of rational dynamics and some hyperbolic PDE systems. Frameworks of characteristic properties of automata groups are inherited to the corresponding rational or PDE dynamics. As an application we study the Burnside problem in group theory and translate the property as the infinite quasi-recursiveness in rational dynamics. automata groups; tropical geometry; dynamical scale transform; asymptotic comparison of PDE Kato, T., Automata in groups and dynamics and tropical geometry, J. Geom. Anal., 24, 901-987, (2014) Geometric theory, characteristics, transformations in context of PDEs, Tropical geometry, Groups acting on trees, Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.), Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), Growth, boundedness, comparison of solutions to difference equations Automata in groups and dynamics and induced systems of PDE in tropical geometry | 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. Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Mirror symmetry (algebro-geometric aspects), , Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Proceedings of conferences of miscellaneous specific interest Homological mirror symmetry and tropical geometry. Based on the workshop on mirror symmetry and tropical geometry, Cetraro, Italy, July 2--8, 2011 | 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 the behaviour of the notion of ``sub-adjoint to a projective variety'' with respect to general hyperplane sections. As an application we show that the two classical definitions of sub-adjoint hypersurface given respectively by Enriques and Zariski are equivalent. subadjoint ideals; hyperplane sections; sub-adjoint hypersurface Varieties and morphisms, Divisors, linear systems, invertible sheaves, Other special types of modules and ideals in commutative rings, Hypersurfaces and algebraic geometry Subadjoint ideals and hyperplane sections | 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 discrete logarithm problem in the group of rational points of an elliptic curve over a finite field is considered to be harder than its traditional multiplicative counterpart. However, there are a few cases where the elliptic problem can can be reduced to the multiplicative one. It was pointed out by \textit{A. I. Menezes}, \textit{T. Okamoto} and \textit{S. A. Vanstone} [IEEE Trans. Inf. Theory 39, 1639--1646 (1993; Zbl 0801.94011)], that this is the case for supersingular curves. Their method exploits the Weil-pairing. Later \textit{G. Frey} and \textit{H.-G. Rück} [Math. Comput. 62, 865--874 (1994; Zbl 0813.14045)] employed the Tate-pairing in certain other cases.
In the present paper the authors discuss practical implementations of both algorithms and compare their running times. elliptic curves; cryptosystems; discrete logarithm; group of rational points; elliptic curve over a finite field; practical implementations; algorithms; running times R. Harasawa, J. Shikata, J. Suzuki, H. Imai, Comparing the MOV and FR reductions in elliptic curve cryptography, in: Advances in Cryptology--Eurocrypt '99, Lecture Notes in Computer Science, Vol. 1592, Springer, Berlin, 1999, pp. 190--205. Cryptography, Applications to coding theory and cryptography of arithmetic geometry Comparing the MOV and FR reductions in elliptic curve cryptography | 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 gives an intriguing approach to the Jacobian conjecture, that if \(f\) is a polynomial mapping from \(C^n\) to itself whose Jacobian matrix has determinant 1, then \(f\) is bijective with polynomial inverse.
\textit{L. M. Druzkowski} [Math. Ann. 264, 303-313 (1983; Zbl 0504.13006)] reduced the problem to special cubic-linear maps, and the author reduces its negation to the problem of whether there exist \(m \times m\) complex matrices \(A\) such that the bilinear function
\[
{\mathcal B} (A) (x,y) : = 3 [\text{diag} (Ax)] [\text{diag} (Ay)]A\tag{1}
\]
is nilpotent for all \(x\) in \(\mathbb{C}^m\); (2) there exist distinct vectors \(x,y\) in \(C^m\) such that \({\mathcal B} (A) (x,y) (x - y) = x - y\). An example satisfying (1) is given and these concepts are also related to the Marcus-Yamabe conjecture [cf. \textit{G. H. Meisters}, Rocky Mt. J. Math. 12, 679-705 (1982; Zbl 0523.34050)]. bad matrix; admissible matrix; Jacobian conjecture; Jacobian matrix; determinant; polynomial inverse; Marcus-Yamabe conjecture Meisters, G. M.: Wanted: a bad matrix. Am. math. Monthly 102, 546-550 (1995) Hermitian, skew-Hermitian, and related matrices, Birational automorphisms, Cremona group and generalizations Wanted: A bad matrix | 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 theorem of Sard type for semialgebraic set-valued mappings whose graphs have dimension no larger than that of their range space: the inverse of such a mapping admits a single-valued analytic localization around any pair in the graph, for a generic value parameter. This simple result yields a transparent and unified treatment of generic properties of semialgebraic optimization problems: ``typical'' semialgebraic problems have finitely many critical points, around each of which they admit a unique ``active manifold'' (analogue of an active set in nonlinear optimization); moreover, such critical points satisfy strict complementarity and second-order sufficient conditions for optimality are indeed necessary. semialgebraic set-valued mappings; Sard-type theorem; subdifferential; strong regularity; metric regularity; identifiable manifold; active set; quadratic growth D. Drusvyatskiy, A. D. Ioffe, and A. S. Lewis, \textit{Generic minimizing behavior in semialgebraic optimization}, SIAM J. Optim., 26 (2016), pp. 513--534. Set-valued and variational analysis, Nonsmooth analysis, Sensitivity, stability, well-posedness, Sensitivity, stability, parametric optimization, Semialgebraic sets and related spaces, Semi-analytic sets, subanalytic sets, and generalizations Generic minimizing behavior in semialgebraic optimization | 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 tight connections between moduli spaces in algebraic and tropical geometry have received lots of attention in recent years. This paper unites and extends all the existing methods in this area, attaining a relation between moduli spaces of log-stable rational maps to toric varieties and their tropical counterparts, which builds on toroidal geometry and non-Archimedean analytic geometry. The result has several interesting consequences and offers new perspectives both for algebraic and tropical geometry. For example, the algebraic moduli space of log-stable rational maps to a toric variety can be viewed as a tropical compactification, or toroidal modification. In tropical geometry, correspondence theorems -- which are at the heart of any successful application of tropical methods in enumerative geometry -- can be deduced from the relation between these moduli spaces.
For part II, see [\textit{D. Ranganathan}, Res. Math. Sci. 4, Paper No. 11, 18 p. (2017; Zbl 1401.14131)]. Ranganathan, D., \textit{skeletons of stable maps I: rational curves in toric varieties}, J. Lond. Math. Soc. (2), 95, 804-832, (2017) Rigid analytic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Skeletons of stable maps. I: rational curves in toric 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 paper deals with ascent and descent properties of finite \(CM\)-type. Let \((R,m)\) be a local ring. If the completion \(\widehat R\) has finite \(CM\)-type it is shown that \(R\) has also finite \(CM\)-type. The converse is shown to be true if \(R\) is Cohen-Macaulay and \(\widehat R\) has an isolated singularity. Another result of the paper shows that if \(k\) is a field and \(k^s\) is the separable closure of \(k\), then the hypersurface \(k[[x_0,\dots,x_d]]/(f)\) has finite \(CM\)-type if and only if \(k^s[[x_0,\dots,x_d]]/(f)\) has finite \(CM\)-type. finite representation type; maximal Cohen-Macaulay modules; ascent; descent; separable closure Wiegand R.: Local rings of finite Cohen--Macaulay type. J. Algebra 203(1), 156--168 (1998) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Commutative rings and modules of finite generation or presentation; number of generators, Cohen-Macaulay modules, Hypersurfaces and algebraic geometry Local rings of finite Cohen-Macaulay 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 \(C\) be a reduced, irreducible projective curve with normalization \(\widetilde C\). The generalized Jacobian \(JC\) of \(C\) is an extension of \(J\widetilde C\) by an affine commutative group. \(J(C)\) is an open subset of the compactified Jacobian \(\overline J C\) of \(C\), the points of which correspond to isomorphism classes of rank one torsion free sheaves \(F\) of degree zero. In the paper under review the Euler number of \(\overline J C\) is computed in the case \(C\) is rational and admits only planar singularities. generalized Jacobian; compactified Jacobian; Euler number Fantechi, B.; Göttsche, L.; van Straten, D., Euler number of the compactified Jacobian and multiplicity of rational curves, J. algebraic geom., 8, 1, 115-133, (1999) Jacobians, Prym varieties, Topological properties in algebraic geometry Euler number of the compactified Jacobian and multiplicity of rational 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 0741.00055.]
This is essentially an expository paper. It gives a unitary reorganization of the material contained in several papers by \textit{R. Maggioni} and the author. In this reorganization several proofs have been improved and some unnecessary hypotheses have been removed.
If \(S\subset\mathbb{P}^ 3\) is a smooth cubic surface, a class of linearly equivalent curves of \(S\) is given by a 7-tuple of integers: the purpose of the paper is to give as much information as possible on the curves lying on \(S\) just using these 7-tuples. So, the dimensions of the cohomology groups \(H^ i(D)\) \((i=0,1,2)\) are computed in \(\S2\) for any divisor \(D\subset S\) and the Hilbert function of any curve (:=effective divisor) \(C\subset S\) is determined in \(\S3\). In sections 4 and 5 the graded Betti numbers of a minimal free resolution of the homogeneous ideal \(I(C)\subset k[x_ 0,x_ 1,x_ 2,x_ 3]\) of any curve \(C\subset S\) are determined. Sections 6 and 7 are devoted to the Rao module \(M(C)\) of any curve \(C\subset S\) linearly equivalent to a smooth one. In \(\S6\) the degrees of the elements of a minimal system of generators of \(M(C)\) are determined; in \(\S7\) a first insight into its structure is given: we see that \(M(C)\) has a nested structure.
Finally, appendix 2 contains some remarks on the smooth cubic surfaces containing Eckardt points. curves on cubic surface; Hilbert function; Rao module; Eckardt points Salvatore Giuffrida, Graded Betti numbers and Rao modules of curves lying on a smooth cubic surface in \?³, The Curves Seminar at Queen's, Vol. VIII (Kingston, ON, 1990/1991) Queen's Papers in Pure and Appl. Math., vol. 88, Queen's Univ., Kingston, ON, 1991, pp. Exp. A, 61. Plane and space curves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special surfaces, Projective techniques in algebraic geometry Graded Betti numbers and Rao modules of curves lying on a smooth cubic surface in \(\mathbb{P}{}^ 3\). Appendix 1: Some commutative diagrams. --- Appendix 2: Eckardt 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 In paragraph 1 the following basic notation and terminology is discussed. A monomial curve \(C(n_ 1,n_ 2,n_ 3)\), \(n_ 1< n_ 2< n_ 3\) positive integers with g.c.d. \((n_ 1,n_ 2, n_ 3)=1\), is defined by its generic zero \((s^{n_ 3}, s^{n_ 3- n_ 1} t^{n_ 1}, s^{n_ 3-n_ 2} t^{n_ 2}, t^{n_ 3})\). Its homogeneous ideal of definition \(I(C)\) is in \({ R}= K[x_ 0, \dots,x_ 3]\), \(K\) an algebraically closed field. It is well known that the Hartshorne-Rao module \(M(C)\simeq H^ 1_{\mathfrak m} (\overline {R})\), where \(H^ 1_{\mathfrak m} (\overline {R})\) is the first local cohomology module of \(\overline {R}= R/I({ C})\) with respect to \({\mathfrak m}= (x_ 0,\dots, x_ 3)\), is of finite length. Thus there exists a minimal nonnegative integer \(k= k({ C})= k(C(n_ 1, n_ 2, n_ 3))\) such that \({\mathfrak m}^ k H^ 1_{\mathfrak m} (\overline {R})=0\). \(k= k({ C})= k(C(n_ 1, n_ 2, n_ 3))\) is called the Buchsbaum number of \({ C}= C(n_ 1, n_ 2, n_ 3)\) and \({ C}= C (n_ 1, n_ 2, n_ 3)\) is said to be (strictly) \(k\)-Buchsbaum. For a \(\mathbb{Z}\)-graded module \(M\) one defines \([M]_ n\) to be the elements of degree \(n\) in \(M\). For \(M\) of finite length let \(a(M)= \min\{n\mid [M]_ n\neq 0\}\) and \(e(M)=\max \{n\mid [M]_ n \neq 0\}\). Then the diameter of \(M\) \((\text{diam} (M))\) equals \(e(M)- a(M)+1\). -- The main result of paragraph 2 (upon which almost all else is based) is to show that \(k=k({ C})= k(C (n_ 1, n_ 2, n_ 3))= \text{diam} (H^ 1_{\mathfrak m} (\overline {R}))\), thus \(k(C (n_ 1, n_ 2, n_ 3))\) actually always assumes its upper bound \(\text{diam} (H^ 1_{\mathfrak m} (\overline {R}))\). Paragraph 3 relates \(k(C (n_ 1, n_ 2, n_ 3))\) algorithmically to a minimal binomial generating set of \(I({ C})\). That the Castelnuovo-Mumford regularity of \(\overline {R}\) equals \(e(H^ 1_{\mathfrak m} (\overline {R}))+ 1\), if \(C (n_ 1, n_ 2, n_ 3)\) is not Cohen-Macaulay, is shown in paragraph 4. Paragraph 5 proves that for monomial curves \(C (1,a,d)\), \(k(1,a,d)= a-2\). In the final paragraph 6 it is shown that for a fixed nonnegative integer \(k\), there are only finitely many linkage classes for (strictly) \(k\)-Buchsbaum monomial curves in \(\mathbb{P}^ 3_ K\). Also no two non Cohen-Macaulay curves of type \(C (1,a,d)\) are in the same even linkage class. liaison; monomial curve; Hartshorne-Rao module; Buchsbaum number; linkage H. Bresinsky, F. Curtis, M. Fiorentini, and L. T. Hoa, On the structure of local cohomology modules for monomial curves in \?³_{\?}, Nagoya Math. J. 136 (1994), 81 -- 114. Plane and space curves, Local cohomology and algebraic geometry On the structure of local cohomology modules for monomial curves in \(\mathbb{P}_ K^ 3\) | 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 smallest possible value of the Euler-Poincaré characteristic \(\chi({\mathcal O}_S) \) for a smooth complex surface \(S\) of general type is \(1\). If \(\chi({\mathcal O}_S)=1\) and \(S\) is minimal, then \(1\leq K^2_S\leq 9\). Furthermore, if \(S\) has \(q>0\), the inequality \(K^2_S\geq 2p_g(S)\) [\textit{O. Debarre}, Bull. Soc. Math. Fr. 110, No.~3, 319-342 (1982; Zbl 0543.14026)] implies \(p_g\leq 4\), and if \(p_g=4\), \(S\) is the product of two curves of genus \(2\) [\textit{A. Beauville}, Appendix to the paper by O. Debarre (loc. cit.)].
In the paper under review the minimal complex surfaces of general type with \(p_g=q=3\) are completely classified. Namely it is shown that such a surface is either the symmetric product of a curve of genus \(3\) or a free \({\mathbb{Z}}_2\)-quotient of the product of a curve of genus \(2\) and a curve of genus \(3\). Furthermore a description of the moduli space of surfaces of general type with \(p_g=q=3\) is given and the degree of the bicanonical map is studied. These two types of surfaces had already been described by \textit{F. Catanese, C. Ciliberto} and \textit{M. Mendes Lopes}, Trans. Am. Math. Soc. 350, No.~1, 275-308 (1998; Zbl 0889.14019), who also showed that the first type is the only example with \(K^2_X=6\) and the second type is the only example with a pencil of curves of genus 2.
The present paper is interesting not only because of the complete classification presented but also because of the innovative approach used, which is completely independent of the methods used in the above cited paper. The main result is obtained by a beautiful use of the generic vanishing theorems of Green and Lazarsfeld and Fourier-Mukai transforms, and it is probably one of the first instances of the use of such tools in the study of surfaces of general type. It should be pointed out that almost contemporarily \textit{G. P. Pirola} [Manuscr. Math. 108, No.~2, 163-170 (2002; Zbl 0997.14009)], using methods similar to the ones of Catanese et alii (loc. cit.) also obtained the main classification result. minimal complex surfaces of general type; generic vanishing theorems Hacon, CD; Pardini, R, Surfaces with \(pg = q = 3\), Trans. Am. Math. Soc., 354, 2631-2638, (2002) Surfaces of general type, Families, moduli, classification: algebraic theory Surfaces with \(p_g=q=3\) | 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 subject matter investigated in the paper are arithmetical Buchsbaum curves in \(\mathbb P^ 3_ K\). A local ring \(A\) is a Buchsbaum ring if for every parameter ideal \(\bar x=(x_ 1,...,x_ d)\), \(\ell(A/_{\bar xA})-e_ 0(\bar xA)=i(A)\) is an invariant. Here \(\ell\) and \(e_ 0\) denote length and multiplicity. A curve in \(\mathbb P_ K^ n\) is arithmetically Buchsbaum if its coordinate ring localized at \((X_ 0,...,X_ n)\) is Buchsbaum. Two ideals \(I\) and \(J\) in a local Gorenstein ring \(A\) are algebraically linked by a complete intersection \(\bar x=(x_ 1,...,x_ g)\subseteq I\cap J\) if (i) \(I\) and \(J\) are ideals of pure height \(g\) and (ii) \(J/_{\bar xA}\simeq \Hom_ A(A/_ I,A/_{\bar xA})\), \(I/_{\bar xA}\simeq \Hom_ A(A/_ J,A/_{\bar xA})\). \(I\) and \(J\) are linked (geometrically) by a complete intersection \(\bar x=(x_ 1,...,x_ g)\) if \(I\) and \(J\) have no common component and \(I\cap J=\bar xA\).
The main result of the paper shows that in a Gorenstein ring \(A\), \(\dim A\geq 1\), the Buchsbaum property is preserved under linkage. A projective variety \(V\subseteq\mathbb P^ n_ K\) is said to be ideally the intersection of \(d\) hypersurfaces if there are \(d\) homogeneous elements \(f_ 1,...,f_ d\) in the defining ideal \(I(V)\) of \(V\) and \(I(V)/(f_ 1,...,f_ d)\) is a \(K[X_ 0,...,X_ n]\)-module of finite length.
A result of independent interest proven to facilitate the study of monomial Buchsbaum curves in \(\mathbb P^ 3_ K\) is the following equivalence for curves \(C\) in \(\mathbb P^ 3_ K\):
(i) \(C\) is arithmetically Buchsbaum (not Cohen-Macaulay) and \(C\) is ideally the intersection of three hypersurfaces \(f_ 1\), \(f_ 2\), \(f_ 3\).
(ii) There are homogeneous elements \(f_ 1\), \(f_ 2\), \(f_ 3\), \(f_ 4\) which generate \(I(C)\) minimally, and \(x_ if_ 4\in(f_ 1,f_ 2,f_ 3)\) for \(i=0,1,2,3\).
Finally for monomial curves in \(\mathbb P^ 3_ K\), i.e. curves \(C\) in \(\mathbb P^ 3_ K\) given parametrically by
\[
\{s^ d,s^ bt^{d-b},s^ at^{d-a},t^ d\},\;d>b>a\geq 1,\;\gcd(d,b,a)=1,
\]
seven equivalent conditions are stated and proven for \(C\) to be arithmetically Buchsbaum. liaison; monomial curves; linkage of arithmetical Buchsbaum curves; Gorenstein ring Henrik Bresinsky, Peter Schenzel, and Wolfgang Vogel, On liaison, arithmetical Buchsbaum curves and monomial curves in \?³, J. Algebra 86 (1984), no. 2, 283 -- 301. Special algebraic curves and curves of low genus, Low codimension problems in algebraic geometry, Multiplicity theory and related topics, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) On liaison, arithmetical Buchsbaum curves and monomial curves in \(\mathbb P^3\) | 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 proves the theorem studied on page 86 ff. of his paper in Serdica 13, No.1, 84-91 (1987; Zbl 0634.51013) and in Rev. Roum. Math. Pures Appl. 32, No.8, 677-686 (1987; Zbl 0635.15027). A similar theorem is published in Collect. Math. 33, 125-139 (1982; Zbl 0523.51021) and also in Demonstr. Math. 16, 723-734 (1984; Zbl 0542.14033) and in Math., Rev. Anal. Numér. Théor. Approximation, Math. 26(49), 33-43 (1984; Zbl 0562.51034). linear system of quadrics Projective analytic geometry, Projective techniques in algebraic geometry, Divisors, linear systems, invertible sheaves Sur le rang de la jacobienne des systèmes linéaires de quadriques. (On the rank of the Jacobian of linear systems of quadrics.) | 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{H. Bass, E. H. Connell} and \textit{D. L. Wright} [Invent. Math. 38, 279--299 (1977; Zbl 0371.13007)] proved that any finitely presented locally polynomial algebra in \(n\) variables over an integral domain \(R\) is isomorphic to the symmetric algebra of a finitely generated projective \(R\)-module of rank \(n\). In this paper we prove a corresponding structure theorem for a ring \(A\) which is a locally Laurent polynomial algebra in \(n\) variables over an integral domain \(R\), viz., we show that \(A\)is isomorphic to an \(R\)-algebra of the form \((\mathrm{Sym}_R(Q))[I^{-1}]\), where \(Q\) is a direct sum of \(n\) finitely generated projective \(R\)-modules of rank one and \(I\) is a suitable invertible ideal of the symmetric algebra \(\mathrm{Sym}_R(Q)\). Further, we show that any faithfully flat algebra over a noetherian normal domain \(R\), whose generic and codimension-one fibres are Laurent polynomial algebras in \(n\) variables, is a locally Laurent polynomial algebra in \(n\) variables over \(R\). polynomial algebra; symmetric algebra; Laurent polynomial algebra; codimension-one; fibre ring ----, The structure of a Laurent polynomial fibration in \(n\) variables , J. Algebra 353 (2012), 142-157. Affine fibrations, Polynomial rings and ideals; rings of integer-valued polynomials, Commutative rings and modules of finite generation or presentation; number of generators The structure of a Laurent polynomial fibration in \(n\) 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 In this paper, the author develops a Bernstein-Sato theory for arbitrary ideals in an \(F\)-finite regular ring of positive characteristic, generalizing work of \textit{M. Mustaţă} [J. Algebra 321, No. 1, 128--151 (2009; Zbl 1157.32012); ``Bernstein-Sato polynomials for general ideals vs. principal ideals'', Preprint, \url{arXiv:1906.03086}] and \textit{T. Bitoun} [Sel. Math., New Ser. 24, No. 4, 3501--3528 (2018; Zbl 1423.13048)] in the case of principal ideals.
Let \(R\) be an \(F\)-finite regular ring of positive characteristic and let \(\mathfrak{a}=(f_1,\dots,f_r)\) be an ideal of \(R\). Consider the local cohomology module \(H_{(f_1-t_1,\dots,f_r-t_r)}^r(R[t_1,\dots,t_r])\). It is well-known that this module has a natural module structure over \(D_{R[t_1,\dots,t_r]}\), the ring of differential operators of \(R[t_1,\dots,t_r]\). The author defines a (decreasing) \(V\)-filtration on \(D_{R[t_1,\dots,t_r]}\), and more precisely on each \(D^e_{R[t_1,\dots,t_r]}=End_{R[t_1,\dots,t_r]^{p^e}}(R[t_1,\dots,t_r])\) (it is well-known that the union of these \(D^e\) is \(D\)). Then the author studies the module \(N_{\mathfrak{a}}^e\) which is defined as the quotient of \(V^0D^e_{R[t_1,\dots,t_r]} \cdot \delta\) by \(V^1D^e_{R[t_1,\dots,t_r]} \cdot \delta\), where \(\delta\) is the element \((f_1-t_1)^{-1}\cdots (f_r-t_r)^{-1}\in H_{(f_1-t_1,\dots,f_r-t_r)}^r(R[t_1,\dots,t_r])\), as well as their limit \(N_{\mathfrak{a}}:=\varinjlim_eN_{\mathfrak{a}}^e\). The higher Euler-type operator \(s_{p^m}:=\sum_{|\underline{a}|=p^m}\partial_{\underline{t}}^{[\underline{a}]}\underline{t}^{\underline{a}}\) acts on \(N_{\mathfrak{a}}^e\) for each \(0\leq m\leq e-1\). The key result which relates \(N_{\mathfrak{a}}^e\) with \(F\)-invariants is Theorem 3.11, where it was shown that the multi-eigenspace of \(N_{\mathfrak{a}}^e\) with eigenvalue \(\alpha=(\alpha_0,\dots,\alpha_{e-1})\) under the action \((s_{p^0},\dots, s_{p^{e-1}})\) is a direct sum of the modules in the set \(\{D_R^e\cdot \mathfrak{a}^{|\alpha|+s{p^e}} / D_R^e\cdot \mathfrak{a}^{|\alpha|+s{p^e}+1}\}\) where \(s=0, 1, \dots, r-1\) and \(|\alpha|=\alpha_0+p\alpha_1+\cdots+p^{e-1}\alpha_{e-1}\), and that each such module occurs in the direct sum. Since it is known (re-proved in this paper) that \(D_R^e\cdot \mathfrak{a}= (C_R^e\cdot \mathfrak{a})^{[p^e]}(=I_e(\mathfrak{a})^{[p^e]})\), it follows that the generalized eigenspace with eigenvalue \(\alpha\) is nonzero precisely when \(I_e(\mathfrak{a}^{|\alpha|+s{p^e}})\neq I_e(\mathfrak{a}^{|\alpha|+s{p^e}+1})\) for some \(0\leq s\leq r-1\). The latter is closely related to \(F\)-invariants such as \(F\)-jumping numbers, thus the author is able to obtain a series of results (Theorem 4.7, Theorem 4.12). The author further introduced Bernstein-Sato roots of \(N_{\mathfrak{a}}\) and proved that they are negative rational numbers (Theorem 6.7), and that there is a connection of these Bernstein-Sato roots with the \(F\)-jumping numbers of \(\mathfrak{a}\) (Theorem 6.11). The results obtained in this article can be viewed as analogs of Bernstein-Sato theory, multiplier ideals, and jumping numbers in characteristic zero, it would be interesting to explore the connections among them. \(F\)-jumping numbers; Bernstein-Sato polynomials; test ideals Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Bernstein-Sato theory for arbitrary ideals in positive characteristic | 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 \(X\) be a complex algebraic variety and \(n\in \mathbb{Z}_{\geq 1}\). The set of semisimple local systems of rank \(n\) on \(X\) is parametrized by the \(\mathbb{C}\)-points of a scheme \(M_B(X,n)\), called the Betti moduli space. The topological properties of the space \(X\) are reflected in the structure of \(M_B(X,n)\) and of naturally defined subsets therein, in particular, of the cohomology jump loci. The author extends the notion of absolute subsets of Betti moduli spaces of smooth algebraic varieties to the case of normal varieties. He proves that twisted cohomology jump loci in rank one over a normal variety are a finite union of translated subtori and shows that the same holds for jump loci twisted by a unitary local system in the case where the underlying variety \(X\) is projective with \(H^1(X,\mathbb{Q})\) pure of weight one. Finally, he studies the interaction of these loci with Hodge theoretic data naturally associated to the representation variety of fundamental groups of smooth projective varieties. cohomology jump loci; absolute sets; Riemann-Hilbert correspondence Homotopy theory and fundamental groups in algebraic geometry, Topological properties in algebraic geometry Cohomology jump loci and absolute sets for singular 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 main part of the paper is a concise exposition of the theory of graduated orders on the space of square matrices with entries in a given valued field. It is based on previous work of Plesken and Zassenhausen, as described in [\textit{W. Plesken}, Group rings of finite groups over \(p\)-adic integers. Berlin etc.: Springer-Verlag (1983; Zbl 0537.20002)]. In addition, the authors provide some further results, which in particular establish connections to tropical geometry and Bruhat-Tits buildings, as well as many detailed examples. graduated orders; polytropes; tropical geometry Foundations of tropical geometry and relations with algebra, Orders in separable algebras, Groups with a \(BN\)-pair; buildings Orders and polytropes: matrix algebras from valuations | 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 To any connected reductive group \(G\) over a non-archimedean local field \(F\) and to any maximal torus \(T\) of \(G\), we attach a family of extended affine Deligne-Lusztig varieties (and families of torsors over them) over the residue field of \(F\). This construction generalizes affine Deligne-Lusztig varieties of Rapoport, which are attached only to unramified tori of \(G\). Via this construction, we can attach to any maximal torus \(T\) of \(G\) and any character of \(T\) a representation of \(G\). This procedure should conjecturally realize the automorphic induction from \(T\) to \(G\). For \(G = \mathrm{GL}_2\) in the equal characteristic case, we prove that our construction indeed realizes the automorphic induction from at most tamely ramified tori. Moreover, if the torus is purely tamely ramified, then the varieties realizing this correspondence turn out to be (quite complicate) combinatorial objects: they are zero-dimensional and reduced, i.e., just disjoint unions of points. affine Deligne-Lusztig variety; automorphic induction; local Langlands correspondence; supercuspidal representations Langlands-Weil conjectures, nonabelian class field theory, Grassmannians, Schubert varieties, flag manifolds, Representation-theoretic methods; automorphic representations over local and global fields Ramified automorphic induction and zero-dimensional affine Deligne-Lusztig 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 A convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few years later, \textit{G. Blekherman} [J. Am. Math. Soc. 25, No. 3, 617--635 (2012; Zbl 1258.14067)] answered the question in the negative by showing through volume arguments that for high enough number of variables, there must be convex forms of degree as low as 4 that are not sums of squares. Remarkably, no examples are known to date. In this paper, we show that all convex forms in 4 variables and of degree 4 are sums of squares. We also show that if a conjecture of Blekherman related to the so-called Cayley-Bacharach relations is true, then the same statement holds for convex forms in 3 variables and of degree 6. These are the two minimal cases where one would have any hope of seeing convex forms that are not sums of squares (due to known obstructions). A main ingredient of the proof is the derivation of certain ``generalized Cauchy-Schwarz inequalities'' which could be of independent interest. convex polynomials; sum of squares of polynomials; Cauchy-Schwarz inequality Semialgebraic sets and related spaces, Projective techniques in algebraic geometry, Convex sets in \(n\) dimensions (including convex hypersurfaces), Inequalities and extremum problems involving convexity in convex geometry On sum of squares representation of convex forms and generalized Cauchy-Schwarz inequalities | 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 reductive group defined over an algebraically closed field of characteristic 0 such that the Dynkin diagram of \(G\) is the disjoint union of diagrams of types \(G_2, F_4, E_6, E_7, E_8\). We show that the degree 3 unramified cohomology of the classifying space of \(G\) is trivial. In particular, combined with articles by Merkurjev [11] and the author [1], this completes the computations of degree 3 unramified cohomology and reductive invariants for all split semisimple groups of a homogeneous Dynkin type. Cohomology theory for linear algebraic groups, Exceptional groups, Galois cohomology of linear algebraic groups, Galois cohomology, Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations Degree 3 unramified cohomology of classifying spaces for exceptional groups | 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 generic Painlevé equations \(P_I\) -- \(P_{VI}\), the algebraic independence conjecture states that if \(y_1\), \(\ldots\), \(y_n\) are distinct solutions of one of these equations viewed as meromorphic functions on some disc, then \(tr\cdot deg_{\mathbb{C}(t)}\mathbb{C}(t)(y_1,y'_1,\ldots ,y_n,y'_n)=2n\).
The conjecture is proved by Nishioka for \(P_I\) and by Nagloo and Pillay for \(P_{II}\), \(P_{IV}\) and \(P_V\).
In the present paper the question is settled for \(P_{III}\) and \(P_{VI}\). The author also proves that any three distinct solutions of a Riccati equation are algebraically independent over \(\mathbb{C}(t)\), provided that there are no solutions in the algebraic closure of \(\mathbb{C}(t)\). Painlevé equation; Riccati equation; algebraic independence conjecture Relationships between algebraic curves and integrable systems, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Model-theoretic algebra Algebraic independence of generic Painlevé transcendents: \(P_{III}\) and \(P_{VI}\) | 0 |
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