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It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties One studies parabolic bundles on a projective variety \(X\) along a simple normal crossing divisor \(D\). The definition of parabolic bundles is given, starting with the notion of \textit{parabolic sheaf} and defining the local freeness of such sheaves. An essential result is the interpretation of the moderated fundamental group as an inverse limit of fundamental groups of stacks of roots. We note also the result concerning a functor ``à la Riemann-Hilbert'' about an equivalence of tannakian categories between local k-vectorial systems of finite rank on a stack and finite bundles. The main result is the theorem 5.2.2 which asserts that, in zero characteristic, finite parabolic bundles are in bijection with representations of the \textit{étale fundamental group} of \(X\setminus D\). This is used in order to define a \textit{fundamental group scheme}, following Nori's ideas. The connections of the results of this paper and previous literature is also discussed (cf. 1.3). parabolic sheaf; fundamental group; fundamental group scheme; Deligne-Mumford stack; stack of roots; normal crossing divisor Borne, N., Sur LES représentations du groupe fondamental d'une variété privée d'un diviseur à croisements normaux simples, Indiana Univ. Math. J., 58, 1, 137-180, (2009) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Homotopy theory and fundamental groups in algebraic geometry On the representations of the fundamental group of a compact variety with a simple normal crossing divisor | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties In the paper under review, the authors study resolutions of ideals that are associated with arrangements of subspaces. Recall that a subspace arrangement \(\mathcal{V}\) is a finite collection \(V_{1},\dots, V_{n}\) of vector subspaces of a given vector space \(V\) over an infinite field \(\mathbb{F}\). For each \(i \in \{1, \dots, n\}\) let \(\{f_{ij} \, : \, j \in \{1, \dots, d_{i}\}\}\) be an ordered \(\mathbb{F}\)-basis of \(V_{i}\). The arrangement of vectors
\[
\{ f_{ij} \, : \, i \in \{1, .\dots, n\}, j \in \{1, \dots, d_{i}\}\}
\]
is a collection of bases of \(\mathcal{V}\). We made an assumption that for a given \(\mathcal{V} = \{V_{1}, \dots, V_{n}\}\) the collection of bases \(\{f_{ij}\}\) is generic in the sense that for all \(a=(a_{1}, \dots, a_{n})\in \mathbb{N}^{n}\) with \(a_{i}\leq d_{i}\) the dimension of \(W_{a}\) is the largest possible, where
\[
W_{a} = \langle f_{ij} \, : \, i \in \{1, \dots, n\} \text{ and } j \in \{1, \dots, a_{i}\}\rangle,
\]
which clearly depends on the subspace arrangement but also on the collection of bases chosen. Notice that a collection of bases satisfying the above assumption always exists. Here the authors consider the product \(J\) of the ideals \(I_{i}\) generated by \(V_{i}\) in the polynomial ring \(S\), i.e., \(S\) is the symmetric algebra of \(V\). It turned out, as an example, that the Hilbert function of \(J\) is combinatorial invariant which means that it just depends on the rank function
\[
\mathrm{rk}_{\mathcal{V}} : 2^{[n]} \rightarrow \mathbb{N}, \quad A \subset \{1, \dots, n\}, \quad \mathrm{rk}_{\mathcal{V}}(A) = \dim_{\mathbb{F}} \sum_{i \in A} V_{i}.
\]
Attached to the rank function we have a discrete polymatroid
\[
P(\mathcal{V}) = \bigg\{ x \in \mathbb{N}^{n} \, : \, \sum_{i \in A} x_{i} \leq \mathrm{rk}_{\mathcal{V}}(A) \, : \text{ for all } A \subset \{1, \dots, n\}\bigg\},
\]
and its subpolymatroid
\[
P(\mathcal{V})^{*} = \bigg\{x \in \mathbb{N}^{n} \, : \, \sum_{i \in A} x_{i} \leq \mathrm{rk}_{\mathcal{V}}(A)-1 \, : \text{ for all } \emptyset \neq A \subset \{1, \dots, n\}\bigg\}
\]
whose ran function \(\mathrm{rk}_{\mathcal{V}}^{*}\) is obtained by the so-called Dilworth truncation, i.e.,
\[
\mathrm{rk}_{\mathcal{V}}^{*}(A) = \min\bigg\{ \sum_{i=1}^{p}\mathrm{rk}_{\mathcal{V}}(A_{i}) - p \, : \, A_{1}, \dots, A_{p} \text{ is a partition of } A \bigg\}.
\]
The authors show, among others, that the ideal \(J\) has linear quotients (which is much stronger than the property of having a linear resolution) and they provide an explicit formula for the Betti numbers and the projective dimension of \(J\). As it was showed, the Betti numbers \(\beta_{i}(J)\) of \(J\) are given by
\[
\sum_{i\geq 0}\beta_{i}(J)z^{i} = \sum_{i \geq 0}\gamma_{i}(\mathcal{V})(1+z)^{i},
\]
where \(\gamma_{i}(\mathcal{V}) = \#\{ x \in P(\mathcal{V})^{*} \, : |x|=i\}\), and the projective dimension of \(J\) is given by
\[
\mathrm{proj.dim}\, J = \mathrm{rk}_{\mathcal{V}}^{*}(\{1, \dots, n\}) = \min\bigg\{\sum_{i=1}^{p} \mathrm{ rk}_{\mathcal{V}}(A_{i}) - p \, : \, A_{1}, \dots, A_{p} \text{ is a partition of } \{1, \dots, n\} \bigg\}.
\]
Furthermore, the authors show that the associated primes of \(J\) are exactly the ideals of the form \(\sum_{i \in A}I_{i}\), where \(A \subset \{1, \dots, n\}\) such that \(\mathrm{rk}_{\mathcal{V}}(A) < \mathrm{rk}_{\mathcal{V}}(B)\) for all \(A \subsetneq B\) and \(\mathrm{rk}_{\mathcal{V}}^{*}(A) = \mathrm{rk}_{\mathcal{V}}(A)-1\). Finally, the authors show that the minimal free resolution of \(J\) can be realized as a subcomplex of the tensor product of the truncated Koszul complexes associated with generic generators of \(V_{i}\)'s. free resolutions; subspace arrangements Syzygies, resolutions, complexes and commutative rings, Configurations and arrangements of linear subspaces Resolution of ideals associated to subspace arrangements | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties This paper addresses the following problem: given a commutative ring A, what is the structure of the set of ''CI points,'' i.e., those maximal ideals generated by dim(A) elements? When A is finitely generated over an algebraically closed field, the author conjectures that this set is a countable union of closed subsets of Max(A). This conjecture is verified when A is 2-dimensional, or if A is regular of dimension 3, using work of Murthy and Mohan Kumar.
An ideal I of height h is said to be projectively generated if there is a rank h projective module mapping onto I. Clearly having this property is intermediate between actually being a complete intersection and being only locally a complete intersection. Let us call a ring A a ''Murthy ring'' if the following are equivalent for all locally CI ideals I of maximal height: (a) I is a complete intersection; (b) I can be projectively generated, and \([A/I]=0\) holds in \(K_ 0\)(A).
The conjecture is shown to be true for Murthy rings. All 1-dimensional rings are Murthy rings, but even 2-dimensional algebras over the real numbers can fail to be Murthy. The author proves that 2-dimensional rings and regular 3-dimensional rings are Murthy rings if they are finitely generated over an algebraically closed field, verifying the conjecture in this case. Several partial results are given, as well as results about the corresponding set-theoretic conjecture. complete intersection point; Chow group; Hilbert scheme; projectively generated ideal; CI points; maximal ideals; Murthy ring; \(K_ 0\) [W1]C. Weibel: ``Complete intersection points on affine varieties{'' Comm. in Algebra 12 (24) (1984) pp. 3011--3051.} Projective and free modules and ideals in commutative rings, Grothendieck groups, \(K\)-theory and commutative rings, Rational and unirational varieties, Ideals and multiplicative ideal theory in commutative rings, Relevant commutative algebra Complete intersection points on affine varieties | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X_{m,d}\subset \mathbb P^{\binom{m+d}{m}-1}\) be the order \(d\) Veronese embedding of \(\mathbb P^m\). Here, we conjecture that for certain secant varieties \(\sigma_t(X_{m,d})\) a hypersurface of it has points of symmetric tensor rank \(>t\). We look at the corresponding conjecture for curves. symmetric tensor rank; Veronese variety; \(X\)-rank; Veronese embedding Projective techniques in algebraic geometry A conjecture on the rank of non-general points of secant varieties of Veronese embeddings of projective spaces | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties If \(P\) is a polygon in \(\mathbb{R}^n\), there exists \(k\in\mathbb{N}\) such that the line bundle over the projective toric variety associated to \(kP\) is very ample. If \(k_{\min}\) is the minimal integer which satisfies this condition, in this paper we will give a sharp majoration of \(k_{\min}\). line bundle over projective toric variety Toric varieties, Newton polyhedra, Okounkov bodies, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Remark on the ampleness of invertible sheaves in compact projective toric variety | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X\) be a hyperbolic curve over \(\overline{\mathbb{Q}_{p}}\). In [Publ. Res. Inst. Math. Sci. 40, No. 4, 1291--1336 (2004; Zbl 1078.14037)], \textit{A. Tamagawa} proved that for every closed point \(x\) of the stable reduction of \(X\), there exists a finite étale cover \(Y\) of \(X\) and an irreducible component of the stable reduction of \(Y\) that lies over \(x\).
In the paper under review, and under the assumption that \(X\) is a Mumford curve, the result is generalized by replacing the stable model of \(X\) by an arbitrary semistable model \(\mathcal{X}\) (and the stable model of \(Y\) by the minimal semistable model of \(Y\) over \(\mathcal{X}\)). This is realized by a clever construction of \(\mu_{p^n}\)-torsors on the Berkovich analytification \(X^{\mathrm{an}}\) of \(X\) (in order to construct points with positive genera on the corresponding covers, which necessarily correspond to generic points of any of their semistable reductions) using theta functions. As a corollary, the author deduces that every curve over \(\overline{\mathbb{Q}_{p}}\) admits a Zariski-dense open subset on which the same result holds.
In the sequel of the paper, the author studies tempered fundamental groups of Mumford curves. Let us recall that a tempered cover of a Berkovich space is a sort of a mix between a finite étale and a topological cover. The notion has been introduced by \textit{Y. André} [Period mappings and differential equations. From \({\mathbb C}\) to \({\mathbb C}_p\). Tôhoku-Hokkaidô lectures in arithmetic geometry. With appendices: A: Rapid course in \(p\)-adic analysis by F. Kato, B: An overview of the theory of \(p\)-adic unifomization by F. Kato, C: \(p\)-adic symmetric domains and Totaro's theorem by N. Tsuzuki. Tokyo: Mathematical Society of Japan (2003; Zbl 1029.14006)] and [Duke Math. J. 119, No. 1, 1--39 (2003; Zbl 1155.11356)]. The author proves that two Mumford curves over \(\overline{\mathbb{Q}_{p}}\) with isomorphic tempered fundamental groups have homeomorphic associated Berkovich spaces.
It is well-known that the Berkovich space associated to \(X\) may be reconstructed by the graphs of all the semistable reductions. The former result of resolution of nonsingularities then shows that it is actually enough to consider stable models of finite Galois covers up to action of the Galois group. The other essential ingredient is a result of Mochizuki that ensures that the tempered fundamental group is enough to recover the graph of the stable reduction (see [\textit{S. Mochizuki}, Publ. Res. Inst. Math. Sci. 42, No. 1, 221--322 (2006; Zbl 1113.14025)]).
Finally, The author handles the case of punctured Tate curves. Let \(q_{1}, q_{2} \in \overline{\mathbb{Q}_{p}}\) such that \(|q_{1}|, |q_{2}| <1\). If the tempered fundamental groups of \(\mathbb{G}_{m}/q_{1}^{\mathbb{Z}} - \{1\}\) and \(\mathbb{G}_{m}/q_{2}^{\mathbb{Z}} - \{1\}\) are isomorphic, then there exists \(\sigma \in \mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})\) such that \(q_{2} = \sigma(q_{1})\). tempered fundamental group; anabelian geometry; Berkovich spaces; nonsingularities; Mumford curves Rigid analytic geometry, Coverings of curves, fundamental group Resolution of nonsingularities for Mumford curves | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The author computes the shape of the minimal free resolution of the ideals of abelian surfaces in \(\mathbb P^4(\mathbb C)\). Every such surface is of the form \(V(s)\) where \(s\) is a section of the Horrocks-Mumford rank 2 vector bundle \(E\) on \(\mathbb P^4(\mathbb C)\). In addition the minimal resolutions of all \(V(s)\) where \(s\) is a section of \(E(n)\), \(n\ge 1\), are computed. For the abelian surfaces, the resolutions are \(G\)-invariant, where \(G\) is a certain subgroup of the group of symmetries of the Horrocks-Mumford bundle. For example, ignoring the \(G\)-structure, the (sheafified) resolution for \(X=V(s)\), \(s\in \Gamma (E)\) is
\[
0\to 20(-10)\to 20{\mathcal O}(-8)\to 35{\mathcal O}(-7)\to 15{\mathcal O}(-6)\oplus 3{\mathcal O}(-5)\to {\mathcal O}\to {\mathcal O}_ X\to 0.
\]
A minimal resolution of \(\Gamma_*(E)\) as a \(\Gamma_*(\mathcal O_{\mathbb P^4})=\mathbb C[X_ 0,\ldots,X_ 4]\)-graded module is also obtained. syzygy; minimal free resolution of the ideals of abelian surfaces; Horrocks-Mumford rank 2 vector bundle DOI: 10.1515/crll.1988.384.180 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Surfaces and higher-dimensional varieties, Abelian varieties and schemes, Projective techniques in algebraic geometry, Complexes Syzygies of abelian surfaces embedded in \(\mathbb P^4(\mathbb C)\) | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X\) be a projective rationally connected surface with canonical singularities carrying a nonzero reflexive pluri-form, that is, the reflexive hull of \((\Omega_X^1)^{\otimes m}\) has a nonzero global section for some \(X\) positive integer \(m\). We show that any such surface \(X\) can be obtained from a rational ruled surface by a very explicit sequence of blow-ups and blow-downs. Moreover, we interpret the existence of nonzero pluri-forms in terms of semistable reduction. W. Ou, Singular rationally connected surfaces with nonzero pluri-forms, Michigan Math. J. 63 (2014), no. 4, 725-745. Rationally connected varieties, Rational and ruled surfaces Singular rationally connected surfaces with nonzero pluri-forms | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties As in \textit{F. Sakai}'s articles [Am. J. Math. 104, 1233-1241 (1982; Zbl 0512.14022) and Saitama Math. J. 1, 29-35 (1983; Zbl 0564.14005)], a normal Gorenstein surface means a complex normal projective surface with at most Gorenstein singularities; this Gorenstein condition is equivalent to invertibility of the dualizing sheaf \(\omega\) (under other conditions). The author begins with some characterization of numerically effectiveness of \(\omega\otimes L\) (L being an ample line bundle on the normal Gorenstein surface S), then he shows several results including a classification of the pair (S,L), in the case the logarithmic Kodaira dimension of the pair \(\neq -\infty\), in view of the dimension of the image of the map associated to \(\Gamma ((\omega \otimes L)^ N)\) with sufficiently large N. ample divisors; del Pezzo surface; normal Gorenstein surface; Gorenstein singularities; dualizing sheaf; ample line bundle; classification Sommese A.I., Abh. Math. Sem. Univ. Hamburg 55 pp 151-- (1985) Divisors, linear systems, invertible sheaves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities in algebraic geometry Ample divisors on normal Gorenstein surfaces | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The cohomological crepant resolution conjecture of \textit{Y. Ruan} [Contemp. Math. 403, 117--126 (2006; Zbl 1105.14078)] states that if \(f: W \to Z\) is a crepant reolution of an orbifold \(Z\), then the orbifold cohomology ring \(H^*_{\text{CR}}(Z)\) introduced by \textit{W. Chen} and \textit{Y. Ruan} [Commun. Math. Phys. 248, No. 1, 1--31 (2004; Zbl 1063.53091)] is isomorphic to the ring \(H^*_f (W)\) obtained from the ordinary cohomology ring of \(W\) by adding quantum corrections related to curves contracted by \(f\). For a simply connected smooth, projective surface \(X\), the authors prove the conjecture for the Hilbert-Chow morphism \(\rho_n: X^{[n]} \to X^{(n)}\) from the Hilbert scheme of \(n\) points to the symmetric product, extending the previously known special cases \(n=2,3\) [\textit{D. Edidin} et al., Asian J. Math. 7, No. 4, 551--574 (2003; Zbl 1079.14061)] and [\textit{W.-P. Li} and \textit{Z. Qin}, Turk. J. Math. 26, No. 1, 53--68 (2002; Zbl 1054.14068)], \(K_X\) is trivial [\textit{B. Fantechi} and \textit{L. Göttsche}, Duke Math. J. 117, No. 2, 197--227 (2003; Zbl 1086.14046)] and [\textit{M. Lehn} and \textit{C. Sorger}, Invent. Math. 152, No. 2, 305--329 (2003; Zbl 1035.14001)] and \(X\) is toric [\textit{W. K. Cheong}, Math. Ann. 356, No. 1, 45--72 (2013; Zbl 1277.14044)].
The proof uses the axiomatic approach developed by \textit{Z. Qin} and \textit{W. Wang} [Contemp. Math. 310, 233--257 (2002; Zbl 1045.14001)] to reduce the question to checking some relations among Heisenberg operators [\textit{H. Nakajima}, Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] acting on \(\mathbb H_X = \bigoplus_{n=0}^\infty H^*(X^{(n)})\) and on \(\oplus_{n} H^*_{\rho_n} (X^{[n]})\). These relations were known for \(H^*(X^{(n)})\) by work of \textit{Qin and Wang} [loc. cit.] and are proved for \(H^*(X^{[n]})\) when \(X\) is toric. To obtain the general case, the authors are able to reduce to the toric case by proving certain universality structures on the 3-pointed genus-0 Gromov-Witten invariants of \(X^{[n]}\) using the indexing techniques of \textit{J. Li} [Geom. Topology 10, 2117--2171 (2006; Zbl 1140.14012)], \textit{K. Behrend}'s product formula for Gromov-Witten invariants [J. Alg. Geom. 8, 529--541 (1999; Zbl 0938.14032)] and the co-sectional localization techniques of \textit{J. Li} and \textit{W.-P. Li} [Math. Ann. 349, 839--869 (2011; Zbl 1221.14006)] and \textit{Y.-H. Kiem} and \textit{J. Li} [J. Amer. Math. Soc. 26, 1025--1050 (2013; Zbl 1276.14083)]. Ruan' conjecture; orbifold cohomology ring; Hilbert-Chow morphism; Hilbert scheme of points Li, W.-P., Qin, Z.: The cohomological crepant resolution conjecture for the Hilbert-Chow morphisms. arXiv:1201.3094 Parametrization (Chow and Hilbert schemes) The cohomological crepant resolution conjecture for the Hilbert-Chow morphisms | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties In this expository note, we review the recent developments about Maulik and Okounkov's stable bases for the Springer resolution \(T^*(G/B)\). In the cohomology case, we compute the action of the graded affine Hecke algebra on the stable basis, which is used to obtain the localization formulae. We further identify the stable bases with the Chern-Schwartz-MacPherson classes of the Schubert cells. This relation is used to prove the positivity conjecture of Aluffi and Mihalcea. For the \(K\)-theory stable basis, we first compute the action of the affine Hecke algebra on it, which is used to deduce the localization formulae via root polynomial method. Similar as the cohomology case, they are also identified with the motivic Chern classes of the Schubert cells. This identification is used to prove the Bump-Nakasuji-Naruse conjecture about the unramified principal series of the Langlands dual group over non-Archimedean local fields. In the end, we study the wall R-matrices, which relate stable bases for different alcoves. As an application, we give a categorification of the stable bases via the localization of Lie algebras over positive characteristic fields. flag variety; Springer resolution; stable bases; Hecke algebra Grassmannians, Schubert varieties, flag manifolds, Representations of Lie and linear algebraic groups over local fields, Modular Lie (super)algebras, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry Stable bases of the Springer resolution and representation theory | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties \textit{N. Fakhruddin} [Multiplication maps of linear systems on smooth projective toric surfaces, \url{arXiv:math/0208178}] showed that for an ample line bundle \(L\) and a globally generated line bundle \(E\) on a nonsingular projective toric surface, the multiplication map \(H^0(L)\otimes H^0(E)\to H^0(L\otimes E)\) is surjective. We give an algebro-geometric proof of this fact with a view toward a possible generalization to singular toric surfaces or the higher dimensional case. As a consequence, we show that on a nonsingular toric Fano threefold, the multiplication map of complete linear systems of the anti-canonical divisor and a nef divisor is surjective. Ogata, S.: Multiplication maps of complete linear systems on projective toric surfaces. Interdiscip. Inf. Sci. 12(2), 93--107 (2006) Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Fano varieties Multiplication maps of complete linear systems on projective toric surfaces | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The authors propose a rather general definition of an embedding \(\varepsilon\) of a point-line geometry \(\Gamma=(\mathcal{P,L})\) to the set \(\mathcal P_V\) of points of a projective space \(\mathrm{PG}(V)\); they define that \(\varepsilon\) is an injective mapping of \(\mathcal P\) to \(\mathcal P_V\) that satisfies three axioms, the most remarkable one is:
E(1) \; For every line \(\ell\in\mathcal L\) the image \(\varepsilon(\ell):=\{\varepsilon(p)\}_{p\in\ell}\) of \(\ell\) spans a \(d\)-dimensional subspace of \(\mathrm{PG}(V)\).
The authors speak of a locally \(d\)-dimensional embedding of \(\Gamma\) in \(\mathrm{PG}(V)\). This definition includes many different situations considered in literature.
The authors ``focus on a class of \(d\)-embeddings, which they call Grassmann embeddings, where the points of \(\Gamma\) are firstly associated to the lines of a projective geometry \(\mathrm{PG}(V)\), next they are mapped onto the points of \(\text{PG}(V\wedge\,V)\) via the usual projective embedding of the line-Grassmannian of \(\mathrm{PG}(V)\) in \(\mathrm{PG}(V\wedge\,V)\).'' The authors present a tool which allows them to describe polar line-Grassmannians by matrix equations, using the isomorphism of \(V\wedge V\) to the space \(S_n(\mathbb F)\) of skew-symmetric \((n\times n)\)-matrices over \(\mathbb F\). The authors consider ``Grassmann embeddings of a number of generalized quadrangles. They also compare those embeddings with other embeddings, which they call quadratic Veronesean embeddings, obtained by composing a projective embedding \(\varepsilon\) with the usual quadratic Veronesean mapping of the projective space hosting \(\varepsilon\).'' embeddings; polar space; generalized quadrangle; exterior products; tensor products; Grassmann variety; Veronese variety Cardinali, I.; Pasini, A., Embeddings of line-Grassmannians of polar spaces in Grassmann varieties. groups of exceptional type, Coxeter groups and related geometries, Springer Proc. Math. Stat., vol. 82, 75-109, (2014), Springer New Delhi Incidence structures embeddable into projective geometries, Polar geometry, symplectic spaces, orthogonal spaces, Generalized quadrangles and generalized polygons in finite geometry, Exterior algebra, Grassmann algebras, Multilinear algebra, tensor calculus, Varieties and morphisms Embeddings of line-Grassmannians of polar spaces in Grassmann varieties | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The Minimal Resolution Conjecture (MRC) for points in projective space said that a general set of points in \(\mathbb P^n\) should have no redundant free summands in its minimal free resolution (i.e. no summands that occur in more than one of the free modules in the resolution). This was shown in \(\mathbb P^3\) by \textit{E. Ballico} and \textit{A. V. Geramita} [in: Algebraic Geometry, Proc. Conf., Vancouver 1984, CMS Conf. Proc. 6, 1-10 (1986; Zbl 0621.14003)] and in \(\mathbb P^4\) by \textit{Ch. H.
Walter} [Math. Z. 219, No. 2, 231--234 (1995; Zbl 0826.14037)]. On the other hand, it was shown to be false in \(\mathbb P^n\) for \(n \geq 6\), \(n \neq 9\), by \textit{D. Eisenbud, S. Popescu, F.-O. Schreyer} and \textit{Ch. H. Walter} [Duke Math. J. 112, No. 2, 379--395 (2002; Zbl 1035.13008)]. A generalization of this conjecture was made by \textit{G Farkas, M. Mustaţǎ} and \textit{M. Popa} [Ann. Sci. Éc. Norm. Supér. (4) 36, No. 4, 553--581 (2003; Zbl 1063.14031)] for the case of a general set of \(t\) (sufficiently large) points on an arbitrary projective variety \(X\) of dimension \(\geq 1\). A few cases of this conjecture have been shown (both to hold and not to hold, depending on \(X\)!). In this paper the author treats the case when \(X\) is a smooth cubic surface in \(\mathbb P^3\). She does not prove the result for all \(t\) sufficiently large, but she does prove it for large ranges of \(t\). All of the cases that she produces are level. The main tool used in this paper is Gorenstein liaison, thus providing an unexpected application of this beautiful theory. minimal resolution conjecture; cubic surface; Gorenstein liaison; linkage; ghost terms Casanellas, M., The minimal resolution conjecture for points on the cubic surface, Can. J. Math., 61, 29-49, (2009) Syzygies, resolutions, complexes and commutative rings, Linkage, complete intersections and determinantal ideals, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Linkage, Low codimension problems in algebraic geometry The minimal resolution conjecture for points on the cubic surface | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We provide a formula to compute the big Cohen-Macaulay test ideal for triples \(((R, \Delta), \mathfrak{a}^t)\) where \(R\) is a mixed characteristic toric ring and \(\mathfrak{a}\) is a monomial ideal. Of particular interest is that this result is consistent with the formulas for test ideals in positive characteristic and multiplier ideals in characteristic zero. big Cohen-Macaulay; test ideals; commutative algebra; algebraic geometry; mixed characteristic Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Homological conjectures (intersection theorems) in commutative ring theory, Singularities in algebraic geometry, Multiplier ideals, Perfectoid spaces and mixed characteristic Big Cohen-Macaulay test ideals on mixed characteristic toric schemes | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We completely classify the \(\mathbb{Q}\)-factorial terminal toric Fano \(3\)-folds such that the sum of the squared torus invariant prime divisors is non-negative. Toric varieties, Newton polyhedra, Okounkov bodies, Fano varieties, Minimal model program (Mori theory, extremal rays) Terminal toric Fano \(3\)-folds with numerical conditions | 0 |
It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties \textit{N. L. White} [Linear Algebra Appl. 31, 81--91 (1980; Zbl 0458.05022)] conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture for high degrees with respect to the rank. This extends our result [the author and \textit{M. Michałek}, Adv. Math. 259, 1--12 (2014; Zbl 1297.14055)] confirming White's conjecture `up to saturation'. Furthermore, we study degrees of Gröbner bases and Betti tables of the toric ideals of matroids of a fixed rank. matroid; toric ideal; base exchange; generating set; Gröbner basis; Betti table Combinatorial aspects of matroids and geometric lattices, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Toric varieties, Newton polyhedra, Okounkov bodies, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) On the toric ideals of matroids of a fixed rank | 0 |