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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The locus of double points obtained by projecting a variety \(X^n\subset \mathbb{P}^N\) to a hypersurface in \(\mathbb{P}^{n+1}\) moves in a linear system which is shown to be ample if and only if \(X\) is not an isomorphic projection of a Roth variety. Such Roth varieties are shown to exist, and some of their geometric properties are determined. double-point divisor; Roth variety; Castelnuovo variety; secant variety; conductor; projection; linear system Ilic B.: Geometric properties of the double-point divisor. Trans. Am. Math. Soc. 350, 1643--1661 (1998) Projective techniques in algebraic geometry, Divisors, linear systems, invertible sheaves Geometric properties of the double-point divisor | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let $f\in\Bbbk[x,y]$ be a primitive irreducible polynomial where $\Bbbk$ is algebraically closed of characteristic zero. The discriminant of $f$ with respect to $y$, $\Delta_y(f)\in\Bbbk[x]$, can be expressed in terms of the resultant $\mathrm{Res}_y(f,\theta_yf)$. In this paper, the authors give lower bounds of the degree of the discriminant $\deg_x\Delta_y(f)$. With $g$ the geometric genus of the algebraic curve defined by $f$ they show that $2g+d_y-1\leq\deg_x\Delta_y(f)\leq 2d_x(d_y-1)$, where $d_x$ and $d_y$ are the partial degrees of $f$ with respect to $x$ and $y$. Their proof uses the relations between the valuation of the discriminant and the Milnor numbers of the curve along the corresponding critical fiber. They also prove, using the embedding line theorem of \textit{S. S. Abhyankar} and \textit{T.-t. Moh} [J. Reine Angew. Math. 276, 148--166 (1975; Zbl 0332.14004)], that in the case where $f$ is a primitive squarefree polynomial with $r$ irreducible factors then $\deg_x\Delta_y(f)\geq d_y-r$. They show that a monic irreducible polynomial with minimal discriminant with respect to $y$ is also monic with minimal discriminant with respect to $x$. Furthermore, they show that irreducible monic polynomials with minimal discriminant coincide with coordinate polynomials. algebraic plane curve; genus; unicuspidal curve; bivariate polynomials; discriminant; Newton polygon; Abhyankar-Moh's embedding line theorem Plane and space curves, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Class numbers, class groups, discriminants, Solving polynomial systems; resultants, Birational automorphisms, Cremona group and generalizations, Singularities of curves, local rings, Special algebraic curves and curves of low genus Plane curves with minimal discriminant | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors give two parametrized versions of the uniformization theorem of a nonconstant, nonhyperbolic Riemann surface. The first constructs the uniformization map directly in terms of coordinates via classical complex analysis; the second one, which is coordinate independent, works over any complex curve and is obtained by extending Kodaira's theory of the Jacobian fibration to a family of singular algebraic curves constructed via algebraic geometry. double section; dominating map; Riemann surface; uniformization map; Jacobian fibration Buzzard, G; Lu, S, Double sections, dominating maps, and the Jacobian fibration, Am. J. Math., 122, 1061-1084, (2000) Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Complex surface and hypersurface singularities, Riemann surfaces; Weierstrass points; gap sequences, Singularities of surfaces or higher-dimensional varieties Double sections, dominating maps, and the Jacobian fibration | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper we study the Hodge numbers of a branched double covering of a smooth, complete algebraic threefold. The involution on the double covering gives a splitting of the Hodge groups into symmetric and skew-symmetric parts. Since the symmetric part is naturally isomorphic to the corresponding Hodge group of the base we study only the skew-symmetric parts and prove that in many cases it can be computed explicitly. threefold; Hodge groups; double covering Cynk, S.: Cohomologies of a double covering of a non-singular algebraic 3-fold, Math. Z. 240, 731-743 (2002) \(3\)-folds, Coverings in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects) Cohomologies of a double covering of a non-singular algebraic 3-fold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A method for determining the number of real quadratic factors of polynomials with real or complex coefficients is introduced. Theorem 1 says: Given a real polynomial \(p(z)=z^{n+2}+b_1z^{n+1}+\ldots+b_{n+2}\) of degree \(n+2\) where \(b_{n+2}\ne 0\), the number of its distinct real quadratic factors equals the number of distinct real solutions of the system
\[
x_1x^n_2 - b_1x^n_2 + p_1(x_1,x_2) = 0,\quad x_2^{n+1} - b_2x^n_2 + x_1p_1(x_1,x_2)+x_2p_2(x_1,x_2) = 0,
\]
where the polynomials \(p_1\) and \(p_2\) are determined by the recurrence relation \(p_m=b_{m+2}x_2^{n-m}-p_{m+2}x_2 - p_{m+1}x_1\), \(m=0,1,\ldots,n\), with boundary conditions \(p_n=b_{n+2}\), \(p_{n - 1}=b_{n+1}x_2-b_{n+2}x_1\).
A necessary condition for the existence of multiple quadratic factors is given. The existence of a factor of form \(z^2+c\) is characterized as the nonsingularity of a matrix written explicitly in terms of the coefficients \(b_1,\ldots,b_{n+2}\). real quadratic factors of polynomials; multiple quadratic factors Polynomials in real and complex fields: factorization, Elementary questions in algebraic geometry, Real polynomials: location of zeros, Polynomials in real and complex fields: location of zeros (algebraic theorems) On the number of real quadratic factors of polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The famous Severi inequality states that a minimal projective surface of general type \(S\) with maximal Albanese dimension satisfies \(K_S^2\geq 4\chi (\mathcal O_S)\). This inequality has a long history (see the review of [\textit{R. Pardini}, Invent. Math. 159, No. 3, 669--672 (2005; Zbl 1082.14041)] and it was finally proven by \textit{R. Pardini} [Invent. Math. 159, No. 3, 669--672 (2005; Zbl 1082.14041)] in characteristic zero. Later \textit{X. Yuan} and \textit{T. Zhang} [Adv. Math. 259, 89--115 (2014; Zbl 1297.14045)] proved that the result holds in every characteristic.
\textit{M. Á. Barja} et al. [J. Math. Pures Appl. (9) 105, No. 5, 734--743 (2016; Zbl 1346.14102)] and \textit{X. Lu} and \textit{K. Zuo} [Int. Math. Res. Not. 2019, No. 1, 231--248 (2019; Zbl 1430.14079)] showed (with completely different proofs) that in characteristic zero a minimal smooth projective surface of maximal Albanese dimension satisfies \(K_S^2= 4\chi (\mathcal O_S)\) if and only if \(q(X)=2\) and the canonical model of \(X\) is a double cover of \(\mathrm{Alb}(X)\) branched on an ample divisor with at most negligible singularities.
The present paper extends to every characteristic of the ground field this result. The proof uses both ideas of Pardini and of Lu and Zuo and the authors need to establish several difficult technical results with special focus in characteristic 2 where as usual many difficulties arise. algebraic surface of general type; Severi inequality; Severi line; double covers; irregular varieties; maximal Albanese dimension Surfaces of general type, Special surfaces, Coverings in algebraic geometry Surfaces on the Severi line in positive characteristic | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Maximal mediated sets (MMS), introduced by
\textit{B. Reznick} [Math. Ann. 283, No. 3, 431--464 (1989; Zbl 0637.10015)],
are distinguished subsets of lattice points in integral polytopes with even vertices. MMS of Newton polytopes of AGI-forms and nonnegative circuit polynomials determine whether these polynomials are sums of squares.
In this article, we take initial steps in classifying MMS both theoretically and practically. Theoretically, we show that MMS of simplices are isomorphic if and only if the simplices generate the same lattice up to permutations. Furthermore, we generalize a result of
\textit{S. Iliman} and the third author [Res. Math. Sci. 3, Paper No. 9, 35 p. (2016; Zbl 1415.11071)].
Practically, we fully characterize the MMS for all simplices of sufficiently small dimensions and maximal 1-norms. In particular, we experimentally prove a conjecture by Reznick for 2 dimensional simplices up to maximal 1-norm 150 and provide indications on the distribution of the density of MMS. lattice; maximal mediated set; nonnegativity; sum of nonnegative circuit polynomials; sums of squares Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Semialgebraic sets and related spaces, Sums of squares and representations by other particular quadratic forms, Forms over real fields, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Convex programming Initial steps in the classification of maximal mediated sets | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Das zu der Fläche dritter Ordnung \(x^3 + y^3 + z^3 = 1\) gehörige Doppelintegral zweiter Gattung:
\[
\iint \frac { y dx dy}z
\]
besitzt im Sinne \textit{Poincaré}s keine Residuen, hat aber dennoch Perioden vergl. F. d. M. 32, 418, 1901, JFM 32.0418.04). Hieraus ergibt sich die Notwendigkeit, den Begriff des zweidimensionalen Cyklus, wenn man nicht eine einzelne Fläche, sondern eine Klasse sich punktweise entsprechender Flächen berachtet, in gewisser Weise zu verallgemeinern, wobei die Fundamentalpunkte und die Ausnahmekurven bei den birationalen Transformationen er algebraischen Flächen eine wichtige Rolle spielen. Residues of double integrals. Surfaces and higher-dimensional varieties Some remarks on the periods of double integrals and the transformation of algebraic surfaces. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Due to Beilinson, a form of tilting theory can be used to establish a connection between algebraic geometry and representation theory of finite dimensional algebras. The starting example is \(\mathbb X=\mathbb P_1(k)\) over a field \(k\), where the the derived Hom-functor \(\mathbf{R}\text{Hom}(T,-)\) defines an equivalence between the derived categories of \(\text{Qcoh}\mathbb X\) and \(\text{Mod}\Lambda\), \(\Lambda\) the the path algebra of the quiver with two nodes and double arrows, that is, the Kronecker algebra. This follows by considering the tiling sheaf \(T=\mathcal O\oplus\mathcal O(1)\) in \(\text{coh}\mathbb X\) with endomorphism ring \(\Lambda\). More examples exist where a noetherian tilting object \(T\) in a triangulated category \(\mathcal D\) provides an equivalence between \(\mathcal D\) and the derived category of \(\text{End}(T)\), e.g. for Calabi-Yau and cluster categories.
To be able to work through the introduction, the reader will need to know the definition of a noncommutative curve (or noncommutative scheme) in general. Section 2 in the article contains one of the best reviews of these concepts, and the reader will gain by going back to the introduction after reading Section 2:
A noncommutative curve \(\mathbb X\) is a category \(\mathcal H\) of coherent sheaves over \(\mathbb X\). \(\mathcal H\) is small, connected, abelian, and every object in \(\mathcal H\) is noetherian. \(\mathcal H\) is a \(k\)-category with finite-dimensional \(\text{Ext}\) -spaces, and there is an autoequivalence \(\tau\) on \(\mathcal H\) (Auslander-Reiten translation) such that Serre duality holds. Finally, \(\mathcal H\) contains an object of infinite length.
Now, an indecomposable coherent sheaf \(E\) which is not of finite length, is torsion free and is called a vector bundle. Then one can write \(\mathcal H=\mathcal H_+\vee\mathcal H_0\) with \(\mathcal H_+\) the class of vector bundles, and so \(\mathcal H_0=\underset{x\in\mathbb X}\coprod\mathcal U_x\) where \(\mathbb X\) is an index set and every \textit{tube} \(\mathcal U_x\) is a connected uniserial length category.
For this definitions to make sense, the authors demands that the set \(\mathcal X\) consists of infinitely many points, and then \(\mathbb X\) is called a \textit{weighted noncommutative regular projective curve} over \(k\). It follows (in the domestic case) that for all points \(x\in\mathbb X\) there are exactly \(p(x)<\infty\) simple objects in \(\mathcal U_x\), and that for almost all, \(p(x)=1\). The numbers \(p(x)\) with \(p(x)>1\) are called the weights.
Weighted noncommutative regular curves are \textit{noncommutative smooth proper curves} (in the sense of Stafford and van den Bergh), and if \(\mathcal H\) in addition to the other properties also admits a \textit{canonical tilting object}, \(\mathbb X\) is a \textit{noncommutative curve of genus 0}.
The weighted projective lines introduced by \textit{W. Geigle} and \textit{H. Lenzing} [Lect. Notes Math. 1273, 265--297 (1987; Zbl 0651.14006)] provide the basic framework for this article. Their main advantage is that they contains a tilting bundle in the category \(\text{coh}(\mathbb X)\). Here, the corresponding finite-dimensional algebras are the canonical algebras which is a much studied object in representation theory.
The tilting objects are \textit{small} if they are noetherian objects, and their endomorphism rings are finite-dimensional algebras. In general, an object \(T\) in a Grothendieck category \(\mathfrak H\) is called \textit{tilting} if \(T\) generates the objects in \(T^{\perp_1}=\{X\in\mathfrak H|\text{Ext}^1(T,X)=0\}\).
These large tilting objects, occurring frequently, are not necessarily in derived correspondence to finite dimensional algebras, but they are connected to \textit{recollements of triangulated categories}, proving strong relationship between the derived categories involved.
The large tilting models are interesting because of their connection with localization of categories: Let \(R\) be a Dedekind domain. Then the tilting modules over \(R\) are parametrized by the subsets \(V\subseteq\text{Max-Spec} R\), obtained by localizations at sets \(V\) of simple modules: The universal localization \(R\hookrightarrow R_V\) yields the tilting module \(T_V=R_V\oplus R_V/R\), and the set \(V=\emptyset\) corresponds to the only finitely generated tilting module, the regular module \(R\).
For arbitrary tame hereditary algebras, the classification of tilting modules is complicated due to the possible presence of finite dimensional direct summands from non-homogeneous tubes. Infinte dimensional tilting modules are parametrized by pairs \((B,V)\) where \(B\) is called a branch module, and \(V\subseteq\mathbb X\). The tilting module corresponding to \((B,V)\) has finite dimensional part \(B\) and an infinite dimensional part of the form \(T_V\) inside a suitable subcategory.
The main goal of this article, is the classification of large tilting objects in hereditary Grothendieck categories (categories acting like the category of vector bundles). Of particular interest is the category \(\text{Qcoh}\mathbb X\) of quasicoherent sheaves over a weighted noncommutaive regular projective curve \(\mathbb X\) over a field \(k\). Also, it is proved how the known results for tame hereditary algebras extend to the general setting.
Note in the following that the authors uses the different perpendicular classes, e.g. \(T^{\perp_1}=\{\mathcal F|\text{Ext}^1(\mathcal F,T)\}=0\).
For a locally coherent Grothendieck category \(\mathfrak H,\) a class \(\mathcal S\subseteq\in\mathcal H\), \(\mathcal H\) the class if finitely presented objects in \(\mathfrak H\), is called \textit{resolving} if it generates \(\mathfrak H\) and has a specified closure property. The authors prove that if \(\text{pd}(S)\leq 1\) for all \(S\in\mathcal S\), there is a tilting object \(T\in\mathfrak H\) with \(T^{\perp_1}=\mathcal S^{\perp_1}\). This is a first existence theorem, leading up to the main results of the article:
For \(\mathbb X\) a weighted noncommutative regular projective curve and \(\mathfrak H=\text{Qcoh}\mathbb X\), the assignment \(\mathcal S\mapsto\mathcal S^{\perp_1}\) is a bijection between resolving classes \(\mathcal S\in\mathcal H\) and tilting classes \(T^{\perp_1}\) of finite type.
The authors develop a lot of tools to do the announced classification, and these results are interesting by themselves. Also, many techniques and inductive arguments are pinpointed, and of great value in other circumstances. They are use to prove the main content of the article which is the classification of large tilting sheaves:
More or less verbatim: Let \(\mathbb X\) be of tubular type. Then every large tilting sheaf in \(\text{Qcoh}\mathbb X\) has a well defined slope \(w\). If \(w\) is irrational, then there is up to equivalence precisely one tilting sheaf of slope \(w\). If \(w\) is rational or \(\infty\), then the large tilting sheaves of slope \(w\) are classified like in the domestic case.
From the above theorem, the authors consider the various special cases, and ends out with many general, interesting results. The article is very clear, and it gives an excellent introduction to noncommutative geometry and some of its applications. The article proves the necessity of noncommutative geometry by generalising results from commutative geometry, and most important; it gives the connection between representation theory and algebraic geometry. weighted noncommutative regular projective curve; tilting sheaf; resolving class; Prüfer sheaf; noncommutative curve of genus zero; domestic curve; tubular curve; noncommutative elliptic curve; slope of quasicoherent sheaf; Grothendieck category; tube; large sheaf; recollement of triangulated category Angeleri Hügel, L., Kussin, D.: Large tilting sheaves over weighted noncommutative regular projective curves (2016). Preprint arXiv:1508.03833 Noncommutative algebraic geometry, Grothendieck categories, Special algebraic curves and curves of low genus, Elliptic curves, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Large tilting sheaves over weighted noncommutative regular projective curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(K\), \(L\) be regular function fields over a field \(k\) with natural surjective homomorphisms of the absolute Galois groups \(G_K,G_L\to G_k\). The (relative) birational anabelian conjecture considers bijectivity of the mapping of \(\Hom_k(L,K)\) into \(\Hom_{G_k}^{\text{open}}(G_K,G_L)\), the classes of the \(G_k\)-compatible open homomorphisms \(G_K\to G_L\) modulo conjugacy by the elements of \(G_{L\bar k}\).
In [Invent. Math. 138, No. 2, 319--423 (1999; Zbl 0935.14019)], among other important results, \textit{S. Mochizuki} proved the bijectivity of this mapping under the assumption that the base field \(k\) is a sub-\(p\)-adic field, i.e., a subfield of a finitely generated field extension over \(\mathbb Q_p\) (\(p\): a prime number). In this paper, investigated is the pro-\(p\) version where \(G_K\), \(G_L\) are replaced by their canonical quotients obtained as extensions of \(G_k\) by the maximal pro-\(p\) quotients of \(G_{K\bar k}\), \(G_{L\bar k}\) respectively. The argument shows that the pro-\(p\) version can be reduced to the case of the trans.degree\((L/k)=1\) which was also established by Mochizuki for sub-\(p\)-adic base fields \(k\). absolute Galois groups; Grothendieck's anabelian conjecture; function field; birational geometry Local ground fields in algebraic geometry, Separable extensions, Galois theory, Arithmetic ground fields for curves, Coverings of curves, fundamental group, Transcendental field extensions The pro-\(p\) hom-form of the birational anabelian conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(H(x, y)\) be a real polynomial in two variable, and \(w= p(x, y) dx+ Q(x, y) dy\) be a differential 1-form with real polynomial coefficients. Then for any regular value \(t\) of \(H\) the form \(w\) is integrated over each oval (connected compact component) of the level curve \(H(x, y)= t\). The function \(I(H, w, t)= \oint_{\delta(t)} w\), where \(\delta(t)\) is an oval in the level \(H(x, y)= t\), is called a complete Abelian integral. The weakened Hilbert 16th problem is the following: find an upper bound for the number of isolated zeros of the integral \(I\) in terms of degrees of \(H\), \(P\), \(Q\). The main result of the paper is
Theorem: if \(H\) is a \(C^2\) Morse function on \(R^2\), and the principal homogeneous part of \(H\) expands as a product of pairwise different linear factors then the number of ovals yielding the zero value of integral \(I\) grows at most as \(\exp O(H)\deg(w)\) as \(\deg(w)\to +\infty\), where the term \(O(H)\) is dependent on \(H\).
This result essentially improves the previous double exponential estimate obtained by Il'yashenko and Yakovenko. oval; complete Abelian integral; weakened Hilbert 16th problem; double exponential estimate Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Stability theory for smooth dynamical systems, Analytic theory of abelian varieties; abelian integrals and differentials, Elliptic functions and integrals Simple exponential estimate for the number of real zeros of complete Abelian integrals | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simply connected complex semisimple Lie group of rank \(r\) with a fixed Borel subgroup \(B\) and a maximal torus \(H\subset B\). Let \(W=\text{Norm}_G(H)/H\) be the Weyl group of \(G\). The generalized flag manifold \(G/B\) can be decomposed into the disjoint union of Schubert cells \(X^\circ_w=(BwB)/B\), for \(w\in W\). To any weight \(\gamma\) that is \(W\)-conjugate to some fundamental weight of \(G\), one can associate a generalized Plücker coordinate \(p_\gamma\) on \(G/B\). In the case of type \(A_{n-1}\) (i.e., \(G=SL_n)\), the \(p_\gamma\) are the usual Plücker coordinates on the flag manifold. The closure of a Schubert cell \(X^\circ_w\) is the Schubert variety \(X_w\), an irreducible projective subvariety of \(G/B\) that can be described as the set of common zeroes of some collection of generalized Plücker coordinates \(p_\gamma\). It is also known that every Schubert cell \(X^\circ_w\) can be defined by specifying vanishing and/or non-vanishing of some collection of Plücker coordinates. The main two problems studied in this paper are the following.
(1) Describe a given Schubert cell by as small as possible number of equations of the form \(p_\gamma=0\) and inequalities of the form \(p_\gamma\neq 0\).
(2) Suppose a point \(x\in G/B\) is unknown to us, but we have access to an oracle that answers questions of the form: ``\(p_\gamma(x)=0\), true or false?'' How many such questions are needed to determine the Schubert cell \(x\) is in?
The number of equations of the form \(p_\gamma=0\) needed to define a Schubert variety is generally much larger than its codimension. We show that for a certain Schubert variety \(X_w\) in the flag manifold of type \(A_{n-1}\), one needs exponentially many such equations to define it, even though \(\text{codim}(X_w) \leq\dim (G/B)={n\choose 2}\). Given this kind of ``complexity'' of Schubert varieties, it may appear surprising that for the types \(A_r,B_r,C_r\), and \(G_2\), we provide a description of an arbitrary Schubert cell \(X^\circ_w\) that only uses \(\text{codim} (X_w)\) equations of the form \(p_\gamma=0\) and at most \(r\) inequalities of the form \(p_\gamma\neq 0\). For the type \(D\), a description of Schubert cells is slightly more complicated. Our main result regarding (2) is an algorithm that recognizes a Schubert cell \(X^\circ_w\) containing an element \(x\). For the types \(A_r,B_r,C_r\), and \(G_2\), our algorithm ends up examining precisely the same Pücker coordinates of \(x\) that appear in the previous result. In the case of type \(A_{n-1}\), recognizing a cell requires testing the vanishing of at most \({n\choose 2}\) Plücker coordinates. Finally, we discuss the problem of presenting a subset of Plücker coordinates whose vanishing pattern determines which cell a point is in. Schubert variety; flag manifold; Plücker coordinate; Bruhat cell; vanishing pattern S. Fomin and A. Zelevinsky, ''Recognizing Schubert cells,'' preprint math. CO/9807079, July 1998. Grassmannians, Schubert varieties, flag manifolds Recognizing Schubert cells. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has exactly one positive eigenvalue at any point on the positive orthant. This property can be seen as an analog of the Hodge-Riemann relations for Lorentzian polynomials.
Lorentzian polynomials are intimately connected to matroid theory and negative dependence properties. We show that matroids, and more generally \(\text{M}\)-convex sets, are characterized by the Lorentzian property, and develop a theory around Lorentzian polynomials. In particular, we provide a large class of linear operators that preserve the Lorentzian property and prove that Lorentzian measures enjoy several negative dependence properties. We also prove that the class of tropicalized Lorentzian polynomials coincides with the class of \(\text{M}\)-convex functions in the sense of discrete convex analysis. The tropical connection is used to produce Lorentzian polynomials from \(\text{M}\)-convex functions.
We give two applications of the general theory. First, we prove that the homogenized multivariate Tutte polynomial of a matroid is Lorentzian whenever the parameter \(q\) satisfies \(0<q\leq 1\). Consequences are proofs of the strongest Mason's conjecture from 1972 and negative dependence properties of the random cluster model in statistical physics. Second, we prove that the multivariate characteristic polynomial of an \(\text{M}\)-matrix is Lorentzian. This refines a result of Holtz who proved that the coefficients of the characteristic polynomial of an \(\text{M}\)-matrix form an ultra log-concave sequence. Lorentzian polynomials; stable polynomials; log-concavity; matroids; M-convexity; tropicalization Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Combinatorial aspects of tropical varieties, Combinatorial inequalities, Combinatorial aspects of algebraic geometry, Combinatorial aspects of matroids and geometric lattices Lorentzian polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let $\mathrm{Poly}_{3}\simeq A_{3}$ be the space of cubic polynomials defined by \[ P_{c,a}(z)=\frac{1}{3}z^{3}-\frac{c}{2}z^{2}+a^{3}, \] which is a branched cover of the parameter space of cubic polynomials with marked critical points. The critical points of $P_{c,a}$ are given by $ c_{0}:=c$ and $c_{1}:=0$. \par The main result of the paper under review is the following: \par Theorem A. An irreducible curve $C$ in the space $\mathrm{Poly}_{3}$ contains an infinite collection of post-critically finite polynomials if and only if one of the following holds. \par 1. One of the two critical points is persistently pre-periodic on $C$, that is, there exist integers $m>0$ and $k\geq 0$ such that: $ P_{c,a}^{m+k}(c_{0})=P_{c,a}^{k}(c_{0})$ or $ P_{c,a}^{m+k}(c_{1})=P_{c,a}^{k}(c_{1})$ for all $(c,a)\in C$. \par 2. There is a persistent collision of the two critical orbits on $C$, that is, there exist $(m,k)\in\mathbb{N}^{2}\backslash \{(1,1)\}$ such that $P_{c,a}^{m}(c_{1})=P_{c,a}^{k}(c_{0})$ for all $(c,a)\in C$. \par 3. The curve $C$ is given by the equation $\{(c,a)$, $ 12a^{3}-c^{3}-6c=0\}$, and coincides with the set of cubic polynomials having a non-trivial symmetry, that is, the set of parameters $(c,a)$ for which $Q_{c}(z):=-z+c$ commutes with $P_{c,a}$. \par Then considering $\mathrm{Per}_{m}(\lambda )$ -- the algebraic curve consisting of those cubic polynomials that admit an orbit of period $m$ and multiplier $ \lambda $, the authors give a characterization of those $\mathrm{Per}_{m}(\lambda )$ that contain infinitely many post -- critically finite (PCF) polynomials. cubic polynomials; special curve; irreducible curve; critical points; post - critically finite polynomials. Plane and space curves Classification of special curves in the space of cubic polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{J. W. Milnor} [Proc. Am. Math. Soc. 15, 275-280 (1964; Zbl 0123.38302)] and \textit{R. Thom} [in: Differ. Combinat. Topology, Sympos. Marston Morse, Princeton, 255-265 (1965; Zbl 0137.42503)] (independently) proved an estimate on the sum of the Betti numbers (relative to an arbitrary field of coefficients) of the set of zeros of polynomials of degree at most \(k>0\) in \(\mathbb{R}^n\), essentially \(C_nk^n\). Most applications of the theorem of Milnor and Thom are to the implied estimate on the number of connected components. Our purpose of this article is to give a proof (following Milnor's methods) of the estimate on the number of connected components that uses only advanced calculus, elementary topology and Sard's theorem (the special cases of Sard's theorem that are used will also be sketeched in this article) that should be accessible to mathematicians and computer scientists who are not experts in algebraic topology. Another is that the proof of lemma 1 in Milnor's paper cited above, left quite a bit to the reader. The first two sections of this article are devoted to an elementary proof of this lemma. Sections 7,8,9 involve more algebraic geometry and constitute whatever is new in this paper. Betti numbers; zeros of polynomials; number of connected components Wallach, NR, On a theorem of Milnor and thom, No. 20, 331-348, (1996), Boston Enumerative problems (combinatorial problems) in algebraic geometry, Topology of real algebraic varieties On a theorem of Milnor and Thom | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the projective space \(\mathrm{PG}(rt-1,q)\) and let \(S\) be a Desarguesian \((t-1)\) spread of the space, \(\Pi\) an \(m\)-dimensional subspace and \(\Lambda\) the linear set consisting of the elements of the spread with non-empty intersection with \(\Pi\), where a linear set is a set of points defined by an additive subgroup of the ambient vector space. The Plücker map is an embedding of the set of subspaces of a vector space with a given dimension into the projective space. The authors describe the image under this embedding of the elements of \(\Lambda\) and they show that it is an \(m\)-dimensional variety, a projection of a Veronese variety of dimension \(m\) and degree \(t\) and that it is a suitable linear section of the Plücker embedding of the elements of \(S\) . Grassmannian; linear set; Desarguesian spread; Schubert variety Desarguesian and Pappian geometries, Combinatorial aspects of finite geometries, Grassmannians, Schubert varieties, flag manifolds On some subvarieties of the Grassmann variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Solutions of an inhomogeneous two-dimensional second-order linear partial differential equation lead to the construction of several types of complex algebraic threefolds. The solutions are related to a one-parameter family of polynomials associated with configurations of real lines in the plane. This way hypersurfaces with many simple singularities (of type \(A\) and \(D\)) can be produced. The construction of Calabi-Yau threefolds is based on special types of line configurations. Mathematica and Singular are used as computing tools. algebraic varieties; singularities; multivariate polynomials Escudero, JG, Threefolds from solutions of a partial differential equation, Exp. Math., 26, 189-196, (2017) \(3\)-folds, Singularities of surfaces or higher-dimensional varieties, Second-order hyperbolic equations Threefolds from solutions of a partial differential equation | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper develops an approach to computing weighted Poincaré polynomials of some twisted wild character varieties from the refined Chern-Simons invariants of torus knots based on a chain of conjectural dualities motivated by string theory (large \(N\) duality, spectral correspondence for Higgs bundles and non-abelian Hodge correspondence).
Consider meromorphic Higgs bundles over a smooth projective curve \(C\) with the polar divisor of the form \(n(p_1+\dots+p_m)\), where \(p_i\) are marked points. Suppose the rank of the bundle factors as \(N=rl\), and the bundle's trivialization over the \(n\)-th order neighborhood of each marked point identifies the Higgs field with the differential of a particular \(\mathfrak{gl}(N,\mathbb{C})\) valued Laurent polynomial. Denote \(\mathcal{H}_{n,l,r}\) the moduli space of stable rank \(1\) Higgs bundles of this sort, twisted in the sense that the Laurent polynomials do not take values in the Cartan subalgebra of \(\mathfrak{gl}(N,\mathbb{C})\). On the one hand, the non-abelian Hodge correspondence relates it to a twisted wild character variety, which is the moduli space of twisted Stokes data \(S_{n,l,r}\). On the other hand, the spectral correspondence relates them to pure dimension \(1\) sheaves on a holomorphic symplectic spectral surface \(S\). The use of this latter correspondence in this context is one of the main novelties of the paper.
Following \textit{M. Kontsevich} and \textit{Y. Soibelman} [``Stability structures, Donaldson-Thomas invariants and cluster transformations'', Preprint, \url{arXiv:0811.2435}], the spectral surface is constructed as the complement of the anti-canonical divisor in the successive blow-up of \(K_C\big(n(p_1+\dots+p_m)\big)\), and the correspondence for closed points is established via local computations. The \(P=W\) conjecture of \textit{M. A. A. De Cataldo} et al. [Ann. Math. (2) 175, No. 3, 1329--1407 (2012; Zbl 1375.14047)] then identifies the weighted Poincaré polynomials of \(S_{n,l,r}\) with the perverse Poincaré polynomials of \(\mathcal{H}_{n,l,r}\). In turn, the refined Gopakumar-Vafa expansion yields explicit predictions for them when the theory of stable pairs for \(K_S\) can be explicitly computed.
When \(C\) is a projective line with a single marked point there is a torus action on \(S\) that lifts to a torus action on \(K_S\), and localizes the theory of stable pairs to a compact torus-invariant curve \(\Sigma\) on \(S\). The stable pairs invariants can then be computed via a refined colored generalization of the Shende-Okounkov-Rasmussen conjecture from the refined Chern-Simons invariants of \(\big(l,(n-2)l-1\big)\) torus knots. Explicit computations are performed for \(n=4,l=2, r\geq1\) and \(n=5,l=r=2\), when \(\Sigma\) is a cuspidal elliptic curve and one can use the Fourier-Mukai transform.
For the unrefined calculations \(C\) is taken to be a higher genus curve with any number \(m\) of marked points, but the same \(n,l,r\) at each. The resulting unrefined versions of the weighted Poincaré polynomials, the \(E\)-polynomials, are colored generalizations of those studied by \textit{T. Hausel} et al. [J. Eur. Math. Soc. (JEMS) 21, No. 10, 2995--3052 (2019; Zbl 1440.14234)]. The authors match the respective predictions for \(0\leq g\leq5\), \(1\leq m\leq3\), \(2\leq l\leq3\), and \(4\leq n\leq6\). twisted Higgs bundles; twisted wild character varieties; large N duality; spectral surface; non-Abelian Hodge correspondence; perverse Poincare polynomials; weighted Poincare polynomials; P=W conjecture; refined Gopakumar-Vafa expansion; torus knots; refined Chern-Simons invariants; Shende-Okounkov-Rasmussen conjecture; E-polynomials Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Knot polynomials, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Twisted spectral correspondence and torus knots | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We revisit Heisenberg indeterminacy principle in the light of the Galois-Grothendieck theory for the case of finite abelian Galois extensions. In this restricted framework, the Galois-Grothendieck duality between finite \(K\)-algebras split by a Galois extension \(L\) and finite \(\mathrm{Gal}(L{:}K)\)-sets can be reformulated as a Pontryagin duality between two abelian groups. We define a \textit{Galoisian quantum model} in which the Heisenberg indeterminacy principle (formulated in terms of the notion of \textit{entropic indeterminacy}) can be understood as a manifestation of a Galoisian duality: the larger the group of automorphisms \(H\subseteq G\) of the states in a \(G\)-set \(\mathcal {O}\simeq G/H\), the smaller the ``conjugate'' algebra of observables that can be consistently evaluated on such states. Finally, we argue that states endowed with a group of automorphisms \(H\) can be interpreted as \textit{squeezed coherent states}, i.e. as states that minimize the Heisenberg indeterminacy relations. Galois-Grothendieck theory; quantum mechanics; Heisenberg indeterminacy principle; symmetries-invariants Page, J., & Catren, G. (2014). Towards a Galoisian interpretation of Heisenberg indeterminacy principle. \textit{Foundations of Physics}. 10.1007/s10701-014-9812-2. Commutation relations and statistics as related to quantum mechanics (general), General and philosophical questions in quantum theory, Étale and other Grothendieck topologies and (co)homologies, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Physics Towards a Galoisian lnterpretation of Heisenberg lndeterminacy principle | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For any positive integers \(n \geq 3\), \(r \geq 1\) we present formulae for the number of irreducible polynomials of degree \(n\) over the finite field \(\mathbb{F}_{2^r}\) where the coefficients of \(x^{n - 1}\), \(x^{n - 2}\) and \(x^{n - 3}\) are zero. Our proofs involve counting the number of points on certain algebraic curves over finite fields, a technique which arose from Fourier-analysing the known formulae for the \(\mathbb{F}_2\) base field cases, reverse-engineering an economical new proof and then extending it. This approach gives rise to fibre products of supersingular curves and makes explicit why the formulae have period 24 in \(n\). supersingular curves; irreducible polynomials; prescribed coefficients; binary fields; characteristic polynomial of Frobenius Ahmadi, Omran; Göloğlu, Faruk; Granger, Robert; McGuire, Gary; Yilmaz, Emrah Sercan, Fibre products of supersingular curves and the enumeration of irreducible polynomials with prescribed coefficients, Finite Fields Appl., 42, 128-164, (2016) Computational aspects of field theory and polynomials, Arithmetic ground fields for curves Fibre products of supersingular curves and the enumeration of irreducible polynomials with prescribed coefficients | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that the set of all compatible preorders on the Picard group of a commutative ring forms a lattice under given operations. Also we discuss the Grothendieck group of a semigroup. compatible preorders; Picard group of a commutative ring; lattice; Grothendieck group of a semigroup Ordered groups, Semigroups, Picard groups, Class groups, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Grothendieck groups (category-theoretic aspects) Lattices of compatible preorders on Picard groups of commutative rings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Following the ideas of Castelnuovo and Enriques, we classify the birational equivalence classes of double planes which are rational or ruled surfaces. In order to do this, we prove that the vanishing of the m-adjoint linear system to the branch curve of the canonical resolution of a double plane, for \(m\geq 2\), is a necessary and sufficient condition for the ruledness of the double plane. double planes; rational surfaces; Cremona transformations A. Calabri, On rational and ruled double planes, Ann. Mat. Pura Appl. 181 (2002), 365--387. Rational and ruled surfaces, Birational automorphisms, Cremona group and generalizations On rational and ruled double planes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials There are several methods in computer algebra to eliminate variables: Gröbner bases, resultants, Ritt-characteristic sets, etc. This paper presents two new elimination procedures for two particular problems: The simultaneous elimination of one variable from several polynomial that is based in a linear-algebra method to compute the degree of the gcd of several univariate polynomials over an integral domain, called \textit{S. Barnett}'s method [Proc. Camb. Philos. Soc. 70, 263-268 (1971; Zbl 0224.15018)], and the simultaneous elimination of several variables in several equations containing a Pham system (a zero-dimensional polynomial system with very good parameter specialization properties) that is based on Hermite's method. The extension of the technique above to more general systems is difficult because it depends on the existence of the universal base for a quotient ring. elimination; greatest common divisor of polynomials; Hermite's method; real algebraic geometry; computer aided geometric design Gonzalez-Vega, L.; Gonzalez-Campos, N.: Simultaneous elimination by using several tools from real algebraic geometry. J. symb. Comput. 28, 89-103 (1999) Effectivity, complexity and computational aspects of algebraic geometry, Numerical computation of solutions to systems of equations, Real algebraic and real-analytic geometry, Polynomial rings and ideals; rings of integer-valued polynomials, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Symbolic computation and algebraic computation, Computer science aspects of computer-aided design Simultaneous elimination by using several tools from real algebraic geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a complex semisimple algebraic group. In [Invent. Math. 166, No. 1, 185--228 (2006; Zbl 1106.14037)], \textit{P. Belkale} and \textit{S. Kumar} defined a new product \(\odot_0\) on the cohomology group \(\mathrm{H}^*(G/P,\mathbb{C})\) of any projective \(G\)-homogeneous space \(G/P\). Their definition uses the notion of Levi-movability for triples of Schubert varieties in \(G/P\).
In this article, we introduce a family of \(G\)-equivariant subbundles of the tangent bundle of \(G/P\) and the associated filtration of the de Rham complex of \(G/P\) viewed as a manifold. As a consequence one gets a filtration of the ring \(\mathrm{H}^*(G/P,\mathbb{C})\) and proves that \(\odot_0\) is the associated graded product. One of the aims of this more intrinsic construction of \(\odot_0\) is that there is a natural notion of a fundamental class \([Y]_{\odot_0}\in(\mathrm{H}^*(G/P,\mathbb{C}),\odot_0)\) for any irreducible subvariety \(Y\) of \(G/P\).
Given two Schubert classes \(\sigma_u\) and \(\sigma_v\) in \(\mathrm{H}^*(G/P,\mathbb{C})\), we define a subvariety \( \sum _u^v \) of \(G/P\). This variety should play the role of the Richardson variety; more precisely, we conjecture that \([\sum_u^v]_{\odot_0}=\sigma_u\odot_0\sigma_v\). We give some evidence for this conjecture, and prove special cases.
Finally, we use the subbundles of \(TG/P\) to give a geometric characterization of the \(G\)-homogeneous locus of any Schubert subvariety of \(G/P\). Belkale-Kumar Schubert calculus; Kostant's harmonic forms Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Distributions on homogeneous spaces and applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The theory of finite fields has become increasingly important in the last twenty years. On the one hand there are classical algebraic and number theoretic problems related to finite fields and on the other hand finite fields have many modern applications in computer science, coding theory and cryptography. This excellent book surveys the most recent achievements in the theory and applications of finite fields and is not meant as an introduction. (For an introduction see the masterpiece of \textit{R. Lidl} and \textit{H. Niederreiter} [Finite fields, Encyclopedia of Mathematics and Its Applications. 20. Cambridge: Cambridge Univ. Press (1996; Zbl 0866.11069)].) The book is evidently an extension of the author's book ``Computational and algorithmic problems in finite fields'' [Mathematics and Its Applications. Soviet Series. 88. Dordrecht: Kluwer Academic Publishers (1992; Zbl 0780.11064)]. The following table of contents can only give a glance of the importance of the book.
1. Polynomial factorization
2. Finding irreducible and primitive polynomials
3. The distribution of irreducible, primitive and other special polynomials and matrices
4. Bases and computations in finite fields
5. Coding theory and algebraic curves
6. Elliptic curves
7. Recurrence sequences in finite fields and cyclic linear codes
8. Finite fields and discrete mathematics
9. Congruences
10. Some related problems (primality testing, integer factorization, lattice basis reduction, algorithmic algebraic number theory, integer polynomials, algebraic complexity theory).
The book suggests numerous open problems and concludes with more than 3000 references. Consequently, it is essential for each researcher in finite field theory and related areas. finite fields; number theory; algebraic number theory; computer science; coding theory; cryptography; algebraic geometry; discrete mathematics; polynomial factorization; counting points on curves; irreducible polynomials; primitive polynomials; bases; discrete logarithm; polynomial multiplication; algebraic curves; exponential sums; elliptic curves; recurrence sequences; cyclic codes; pseudo-random numbers; permutation polynomials; congruences; integer factorization; primality testing; computational algebraic number theory; algebraic complexity theory; polynomials with integer coefficients I. E. Shparlinski, \textit{Finite Fields: Theory and Computation}, Kluwer Academic Publ., Dordrecht, 1999 (to appear). Finite fields and commutative rings (number-theoretic aspects), Research exposition (monographs, survey articles) pertaining to number theory, Number-theoretic algorithms; complexity, Algebraic coding theory; cryptography (number-theoretic aspects), Cryptography, Geometric methods (including applications of algebraic geometry) applied to coding theory, Polynomials over finite fields, Exponential sums, Arithmetic theory of polynomial rings over finite fields, Random number generation in numerical analysis, Analysis of algorithms and problem complexity, Symbolic computation and algebraic computation, Algebraic number theory computations, Rational points, Curves over finite and local fields, Factorization, Primality Finite fields: theory and computation. The meeting point of number theory, computer science, coding theory and cryptography | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is the third part of my thesis [at the Mathematics Institute of Fudan Univ., P.R. China (1987)]. Roughly speaking, the topics discussed in my thesis are so-called surfaces of general type with \(\chi\) (\({\mathcal O}_ S)=1\) and fibrations of genus two. For this kind of surfaces, we may find a double covering over a ruled surface, say P. Therefore, one can classify them (as we did in part I and II) by giving the invariants of P and classify the branch locus of the corresponding double covering.
Once we know this basic method, we may use certain results of \textit{Xiao}, \textit{Persson} and \textit{Beauville} for double covering. For example, one can easily know that basically, we only need to classify the surfaces with \(q(S)=p_ g(S)=0, 1\) and 2. For example for surfaces with \(q=0\) in our case, P should be a product of two projective lines. In this situation, using a result of \textit{Horikawa}, one can show that there are only a few choices for the branch curves. - Now we may study the singular points on the branch locus. Suppose its non-fiber part are the summation of curves \(C_ j\). We at first determine the types of \(C_ j's\). Then we give the singularities on \(C_ j's\). Finally, we also show the situation for the intersection among \(C_ i\) and \(C_ j\) for \(i\neq j\). In this sense, we classify all the surfaces of general type with \(\chi\) (\({\mathcal O}_ S)=1\) and fibration of genus 2. As a by-product, we also can give the 2-torsion of the algebraic fundamental group of our surface.
Having classified those surfaces, we try to construct them. We pay our most attention to the surfaces with \(p_ g=0\), as they are very interesting. - By our classification, now the self-intersection of the classical sheaf is 1 or 2. Such surfaces do exist by the constructions of \textit{Oort-Peters} and \textit{Xiao} respectively. Now in this part III (under review), we construct another one. We give the exact bipolynomials, which define the curves in the branch locus. By the way, in a paper of \textit{Reid}, there is also constructed another example by our classification. Now by additional work of mine, there are only four types which we do not know if they exist. For this, please see a forthcoming book written by \textit{Xiao} about surfaces with fibrations. classification of surfaces of general type; construction of surfaces of general type; double covering; branch locus Families, moduli, classification: algebraic theory, Coverings in algebraic geometry, Surfaces of general type, Moduli, classification: analytic theory; relations with modular forms, Families, fibrations in algebraic geometry Surfaces of general type with \(\chi({\mathcal O}_ S)=1\) and fibrations of genus two. III | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group \(\mathrm{O}_3\) on the space \(\mathbb{R}[x,y,z]_{2d}\) of ternary forms of even degree \(2d\). The construction relies on two key ingredients: on the one hand, the Slice Lemma allows us to reduce the problem to determining the invariants for the action on a subspace of the finite subgroup \(\mathrm{B}_3\) of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed \(\mathrm{B}_3\)-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the \(\mathrm{B}_3\)-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the \(\mathrm{O}_3\)-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed \(\mathrm{B}_3\)-invariants to determine the \(\mathrm{O}_3\)-orbit locus and provide an algorithm for the inverse problem of finding an element in \(\mathbb{R}[x,y,z]_{2d}\) with prescribed values for its invariants. These computational issues are relevant in brain imaging. computational invariant theory; harmonic polynomials; orthogonal group; slice; rational invariants; diffusion MRI; neuroimaging Actions of groups on commutative rings; invariant theory, Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Geometric invariant theory, Symmetric groups, Representations of finite symmetric groups, Symbolic computation and algebraic computation, Image processing (compression, reconstruction, etc.) in information and communication theory, Biomedical imaging and signal processing Rational invariants of even ternary forms under the orthogonal group | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper an explicit formula is proven for the multiplication of an arbitrary Schubert cycle by a special Schubert cycle in the Chow (or cohomology) ring of the homogeneous spaces \(Sp (2m)/P\) and \(SO (2m + 1)/P\), where \(P\) is a maximal parabolic subgroup. These homogeneous spaces are interpreted as Grassmannians of isotropic subspaces of a fixed dimension in \(2m\)-dimensional (resp. \((2m + 1)\)-dimensional) vector space endowed with a non-degenerate symplectic (resp. orthogonal) form. The method follows an earlier paper by the author [Manuscr. Math. 79, No. 2, 127-151 (1993; Zbl 0789.14041)] and uses the divided difference description of Borel's characteristic map in the basis of Schubert cycles given by \textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand} [Russ. Math. Surv. 28, No. 3, 1-26 (1973); translation from Usp. Mat. Nauk 28, No. 3(171), 3-26 (1973; Zbl 0286.57025)] and \textit{M. Demazure} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)]. This allows one to reformulate the original intersection theory problem into some questions of purely algebro-combinatorial nature. As a by-product one obtains some Giambelli-type formulas for these isotropic Grassmannians. Schubert cycle; isotropic Grassmannians Pragacz, P., Ratajski, J.: A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians. J. Reine Angew. Math. 476, 143--189 (1996) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Homogeneous spaces and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper places itself within the rich framework of the literature regarding quantum cohomology of homogeneous varieties produced by the authors themselves. This new piece of mathematics regards certain \textsl{Quantum Giambelli} formulas for isotropic Grassmannians. If \(V\) is a vector space equipped with a non degenerate bilinear form \(\eta\), one can consider the Grassmannian \(X:=IG(k,V)\) of isotropic subspaces of \(V\) of a fixed dimension \(k\), namely the variety of those points \([W]\in G(k,V)\) such that the restriction of \(\eta\) to \(W\) is trivial (i.e. \(\eta_{|W\times W}=0\)).
If the bilinear form is skew-symmetric, the dimension of \(V\) is even and one is then concerned with symplectic vector spaces. As well known, the cohomology ring \(H^*(X,{\mathbb{Z}})\) is generated as a \({\mathbb{Z}}\)-algebra by certain special Schubert cycles and it is also a well known fact that such cycles generate the quantum cohomology of \(X\) as well. The latter is a deformation of the usual cohomology encoding the Gromov-Witten invariants which count, roughly speaking, numbers of maps of a given degree from the projective line to \(X\). The authors find and prove quantum Giambelli's formulas expressing an arbitrary Schubert class in the small quantum cohomology ring of \(X\) as a polynomial in the special Schubert classes alluded above.
The two main theorems of the article (concerning Giambelli's formulas) are analogous to those proven for the quantum cohomology of the orthogonal and Lagrangian Grassmannians in [J. Algebr. Geom. 12, No. 4, 777--810 (2003; Zbl 1051.53070)] and [Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083)], by \textit{A. Kresch} and \textit{H. Tamvakis}. The proof are however quite different, due to the fact that for non maximal isotropic Grassmannians, the explicit recursion used in the quoted references is no longer available. The latter is replaced, however, by another kind of recursion, neatly steted and proved in Proposition 3.
The reviewed paper is for all people interested in the combinatorial aspects of cohomology theories (quantum, equivariant, quantum-equivariant) of homogeneous varieties. quantum Schubert Calculus; isotropic Grassmannians; Giambelli's Formulas Buch, AS; Kresch, A; Tamvakis, H, Quantum Giambelli formulas for isotropic Grassmannians, Math. Ann., 354, 801-812, (2012) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Quantum Giambelli formulas for isotropic Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We establish one direction of a conjecture by \textit{N. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45-52 (1990; Zbl 0714.14033)] which describes combinatorially the singular locus of a Schubert variety. We prove that the conjectured singular locus is contained in the singular locus. singular locus of a Schubert variety Gasharov, Vesselin, Sufficiency of Lakshmibai-Sandhya singularity conditions for Schubert varieties, Compos. Math., 126, 1, 47-56, (2001), MR 1827861 Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Combinatorics of partially ordered sets Sufficiency of Lakshmibai-Sandhya singularity conditions for Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A tower \(\cdots\rightarrow C_n\rightarrow C_{n-1} \rightarrow \cdots\rightarrow C_1\rightarrow C_0\) of covers of curves \(C_n/\mathbb F_q\) is asymptotically good if \(\lim \#C_n(\mathbb F_q)/g(C_n)>0\), where \(\#C_n(\mathbb F_q)\) is the number of \(\mathbb F_q\)-rational points of \(C_n\) and \(g(C_n)\) is the genus of \(C_n\). Asymptotically good towers are important in coding theory, but then the curves have to be in explicit form.
In this short paper the author answers a question arising from the work of \textit{A. Garcia}, \textit{H. Stichtenoth}, and \textit{M. Thomas} [Finite Fields Appl. 3, 257--274 (1997; Zbl 0946.11029)] on the construction of explicit asymptotically good towers. Garcia et al. obtained a simple explicit example over any non-prime field by applying their general result (Theorem 2.2) which asserts that an asymptotically good tower can be constructed via a polynomial \(f(t)\) with certain properties; they show that such \(f(t)\) exists over any non-prime field, but they cannot find such \(f(t)\) over a prime field.
The author shows (Theorem 2) that over a prime field there does not exist any polynomial \(f(t)\) satisfying the conditions of Theorem 2.2 of Garcia et al., so their construction cannot be used to obtain asymptotically good towers over prime fields. finite fields; polynomials; curves with many points Lenstra, H. W., On a problem of garcia, stichtenoth, and Thomas, Finite Fields Appl., 8, 166-170, (2002) Curves over finite and local fields, Finite ground fields in algebraic geometry, Arithmetic theory of algebraic function fields On a problem of Garcia, Stichtenoth, and Thomas. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Hodge conjecture predicts that the \(\mathbb{Q}\)-linear span of the classes of algebraic subvarieties in the cohomology of a smooth complex projective variety \(X\) is given by the Hodge ring \(\text{Hdg} (X):= \bigoplus_p H^{2p} (X, \mathbb{Q}) \cap H^{p,p} (X,\mathbb{C})\). Grothendieck's standard conjecture A [\textit{A. Grothendieck}, in: Algebr. Geom., Bombay Colloq. 1968, 193-199 (1969; Zbl 0201.23301)]\ states that for a smooth, projective variety \(X\) over \(\mathbb{C}\), the operator \(\Lambda\) of Hodge theory takes algebraic cycles to algebraic cycles.
In this paper we show that the Hodge conjecture is true for all abelian varieties if the analog of standard conjecture A is true for the \(L_2\)-cohomology of Kuga fiber varieties. By this we mean that the Hodge \(*\)-operator on the \(L_2\)-cohomology takes algebraic cycles to algebraic cycles. The linkage between the Hodge conjecture and the standard conjectures is the following conjecture of Grothendieck, which we shall refer to as the invariant cycles conjecture:
Let \(f: A\to V\) be a smooth and proper morphism of smooth quasiprojective varieties over \(\mathbb{C}\). Let \(P\in V\), and \(\Gamma:= \pi_1 (V, P)\). The space of all \(s\in H^0 (V, R^b f_* \mathbb{Q}) \cong H^b (A_P, \mathbb{Q})^\Gamma\), which represent algebraic cycles in \(H^b (A_P, \mathbb{Q})^\Gamma\), is independent of \(P\). Grothendieck standard conjecture; Hodge theory; \(L_ 2\)-cohomology of Kuga fiber varieties; invariant cycles conjecture S. Abdulali, Algebraic cycles in families of abelian varieties,Can. J. Math.,46 (6) (1994), 1121--1134. Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic cycles, Algebraic moduli of abelian varieties, classification Algebraic cycles in families of abelian varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(A\) be a noetherian ring and \(M\) a finitely generated \(A\)-module. If \(A\) is a domain then \textit{H. Rossi} proved in [Rice Univ. Stud. 54, 63--73 (1968; Zbl 0179.40103)] that there is a projective birational morphism \(\pi:X\rightarrow \text{Spec}(A)\) such that \(\pi^*(M)/\text{torsion}(\pi^*(M))\) is flat \({\mathcal O}_X\)-module and \(\pi\) is universal with this flattening property. The aim of this paper is to extend Rossi's result in the case when \(A\) is not reduced. Thus the author had to extend in this frame many notions including the birational morphism. blow-up; birational morphism; Grothendieck flattening; geometrically flat modules Villamayor U., O. E., On flattening of coherent sheaves and of projective morphisms, J. Algebra, 295, 1, 119-140, (2006), MR 2188879 Rational and birational maps, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Global theory and resolution of singularities (algebro-geometric aspects) On flattening of coherent sheaves and of projective morphisms | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials If a polynomial map \(f:{\mathbb C}^n\to {\mathbb C}\) has a nice behaviour at infinity (e.g. it is a ``good polynomial''), then the Milnor fibration at infinity exists; in particular, one can define the Seifert form at infinity \(\Gamma(f)\) associated with \(f\).
In this paper we prove a Sebastiani--Thom type formula. Namely, if \(f:{\mathbb C}^n\to {\mathbb C}\) and \(g:{\mathbb C}^m\to {\mathbb C}\) are ``good'' polynomials, and we define \(h=f\oplus g: {\mathbb C}^{n+m}\to {\mathbb C}\) by \(h(x,y)=f(x)+g(y)\), then \(\Gamma(h)=(-1)^{mn} \Gamma(f)\otimes \Gamma(g)\). This is the global analogue of the local result, proved independently by K. Sakamoto and P. Deligne for isolated hypersurface singularities. good polynomials; Milnor fibrations at infinity András Némethi, On the Seifert form at infinity associated with polynomial maps, J. Math. Soc. Japan 51 (1999), no. 1, 63 -- 70. Milnor fibration; relations with knot theory, Topological properties in algebraic geometry, Geometric invariant theory, Knots and links (in high dimensions) [For the low-dimensional case, see 57M25] On the Seifert form at infinity associated with polynomial maps | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \( z=(z_1,\cdots,z_n)\) and let \( \Delta=\sum_{i=1}^n \frac{\partial^2}{\partial z^2_i}\) be the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is equivalent to what we call the vanishing conjecture:
For any homogeneous polynomial \( P(z)\) of degree \( d=4\), if \( \Delta^m P^m(z)=0\) for all \( m \geq 1\), then \( \Delta^m P^{m+1}(z)=0\) when \( m>>0\), or equivalently, \( \Delta^m P^{m+1}(z)=0\) when \( m> \frac{3}{2}(3^{n-2}-1)\).
It is also shown in this paper that the condition \( \Delta^m P^m(z)=0\) (\( m \geq 1\)) above is equivalent to the condition that \( P(z)\) is Hessian nilpotent, i.e. the Hessian matrix \( \mathrm{Hes}\,P(z)=(\frac{\partial^2 P}{\partial z_i\partial z_j})\) is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen and various results obtained in this paper on Hessian nilpotent polynomials. Some further results on Hessian nilpotent polynomials and the vanishing conjecture above are also derived. deformed inversion pairs; the heat equation; harmonic polynomials Zhao W., Hessian nilpotent polynomials and the Jacobian conjecture, Trans. Amer. Math. Soc., 2007, 359(1), 249--274 Jacobian problem, Harmonic, subharmonic, superharmonic functions in higher dimensions, Biharmonic and polyharmonic equations and functions in higher dimensions, Spherical harmonics, Boundary value problems for higher-order elliptic equations, Functional equations for complex functions Hessian nilpotent polynomials and the Jacobian conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using Dynkin graphs to investigate RDP (rational double points) on surfaces was first done by \textit{T. Urabe} [Invent. Math. 87, No. 3, 549- 572 (1987; Zbl 0612.14035)], where he gave a partially description of the RDP configurations which occur on a quartic surface in \(\mathbb{P}^ 3\). The purpose of this present paper continuous to deal with RDP on a normal octic \(K3\)-surface in \(\mathbb{P}^ 5\). The author proves that starting with any of 19 basic graphs, and performing two of the elementary transformations of the Dynkin graphs, one reaches a configuration that actually occurs on a normal octic \(K3\) surface in \(\mathbb{P}^ 5\), which is in fact a complete intersection of three quadrics. The question of RDP on such a surface can be purely changed into that of lattice theory. Thus the determination of RDP on a normal octic \(K3\) surface is fully dependent on the development of lattice theory. In the current paper, however, there exist configurations on a normal \(K3\) surface, which are not covered by the main theorem. The method of this paper is mainly lattice-theoretic, following the work of V. V. Nikulin. octic \(K3\) surface; rational double points; Dynkin graphs Singularities of surfaces or higher-dimensional varieties, \(K3\) surfaces and Enriques surfaces, Singularities in algebraic geometry Rational double points on a normal octic \(K3\) surface | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0742.00065.]
The author proves some results on Donaldson's polynomials of algebraic surfaces \(S\). First he reviews Donaldson's theorem on positivity of the value of \(\gamma_ k\) on a hyperplane class. Then he proves that the hypersurface in \(\mathbb{P}(H_ 2(S;\mathbb{C}))\) defined by the vanishing of \(\gamma_ k\) has special properties. This is accomplished by considering the intersection of the hypersurface with a plane spanned by (the Poincaré duals of) three elements: one in \(H^{2,0}(S)\), one in \(H^{0,2}(S)\), and one in \(H^{1,1}(S)\). Donaldson's polynomials of algebraic surfaces Tyurin, A.: A slight generalization of the Mehta-Ramanathan theorem. InAlgebraic Geometry, Proceedings Chicago 1989SLNM1479258--272 (1991) Classical real and complex (co)homology in algebraic geometry, Special surfaces, Realizing cycles by submanifolds A slight generalization of the Mehta-Ramanathan theorem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A degree \(d\) smooth space curve \(C\) with \(H^ 0 (\mathbb{P}^ 3, {\mathcal I}_ C (k-1)) = 0\) for some \(k\) is said to be in the range \(B\) if \((k^ 2 + 4k + 6)/3 \leq d \leq k^ 2\). Here the author (with the very restrictive assumption that \(C\) has maximal rank) proves the upper bound for the genus of \(C\) conjectured by Hartshorne and Hirschowitz. The inductive proof uses a sequence of descending double linkages as in previous papers by G. Fløystad and C. Walter. space curve; postulation; liaison; hyperplane section; genus; double linkages Rosario Strano, On the genus of a maximal rank curve in \?³, J. Algebraic Geom. 3 (1994), no. 3, 435 -- 447. Plane and space curves, Linkage On the genus of a maximal rank curve in \(\mathbb{P}^ 3\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let J be a finite-dimensional simple Jordan algebra over an algebraically closed field of characteristic 0 or let J be simple and formally real over \({\mathbb{R}}\). Let \(\lambda\) be the reduced trace of J and r=degree of J. Then the r forms \(x\to \lambda (x^ k)\), \(x\in J\), \(1\leq k\leq r\), are algebraically independent over K and generate the algebra of Aut J- invariant polynomials, where Aut J=automorphism group of J.
The theorem is proven in the same way as the corresponding theorem for Lie algebras [see e.g. \textit{N. Bourbaki}, Groupes et algèbres de Lie (1975; Zbl 0329.17002), Chap. 8, {\S}8.3, Théorème 1]. Also, a similar theorem holds for symmetric spaces of noncompact type [see \textit{S. Helgason}, Differential geometry and symmetric spaces (1962; Zbl 0111.181), Chap. X, {\S} 6.]. Note that in the formally real case (Str(J), Aut(J)), Str(J)=structure group of J is a reductive Riemannian symmetric pair. simple Jordan algebra; formally real; reduced trace; invariant polynomials; automorphism group Associated groups, automorphisms of Jordan algebras, Group actions on varieties or schemes (quotients), Simple, semisimple Jordan algebras Invariant polynomial functions on Jordan algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We develop a theory of residues for arithmetic surfaces, establish the reciprocity law around a point, and use the residue maps to explicitly construct the dualizing sheaf of the surface.
These are generalisations of known results for surfaces over a perfect field.
In an appendix, explicit local ramification theory is used to recover the fact that in the case of a local complete intersection the dualizing and canonical sheaves coincide. residues; reciprocity laws; arithmetic surfaces; Grothendieck duality Morrow, M., An explicit approach to residues on and dualizing sheaves of arithmetic surfaces, New York J. Math., 16, 575-627, (2010) Arithmetic varieties and schemes; Arakelov theory; heights, Ramification and extension theory, Arithmetic ground fields for curves, Local cohomology and algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials An explicit approach to residues on and dualizing sheaves of arithmetic surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author starts by recalling definitions and constructions referred to later in the discussion of noncommutative projective geometry and algebraic aspects of noncommutative tori. Artin, Tate and Van den Bergh have given definitions of noncommutative algebras that should in principle be algebras of functions on some nonsingular noncommutative schemes. The author gives the necessary conditions needed on the graded algebras, and he defines \textit{Artin-Schelter (AS)-regularity}, which is the nonsingularity condition in the noncommutative setting. Also the noncommutative Gorenstein condition is defined in the noncommutative situation. This leads to the definition of a \textit{standard algebra}, containing the essential properties of AS-regular algebras of dimension 3.
The text contains the introductory section about twisted homogeneous coordinate rings which gives a general recipe for constructing noncommutative rings out of a commutative geometric datum, called an abstract triple which is an isomorphism invariant for AS-regular algebras. This construction is completely explicit, and a direct computation in a simple example is given.
A very short, nice introduction to Grothendieck categories is given, and the Gabriel-Rosenberg theorem is explained and exploited: ``Any scheme can be reconstructed from the category of quasi-coherent sheaves on it''. The author explains the idea of a quotient category, and then he gives the model of noncommutative projective geometry after Artin and Zhang: The triple \((QGr(R),R,s)\) is called the projective scheme of \(R\) and is denoted \(\text{Proj}(R)\). Also the characterization of \(\text{Proj}(R)\) given by Artin and Zhang gets a thorough treatment. The final chapter in this article on ``algebraic aspects of noncommutative tori'' is not completely self-contained, but the main idea is treated in an understandable way.
Reviewer's remark: This article is one of the most accessible reviews of noncommutative geometry given, leaving out the hardest proofs and the most general descriptions of the theme. noncommutative projective geometry; Grothendieck categories; \(\chi\)-condition; quotient categories Noncommutative algebraic geometry, Grothendieck categories, Derived functors and satellites Lecture notes on noncommutative algebraic geometry and noncommutative tori | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In his article [in J. Am. Math. Soc. 11, No. 2, 229-259 (1998; Zbl 0904.20033)], \textit{F. Brenti} gave a non-recursive formula for the Kazhdan-Lusztig polynomials of a Coxeter group and proved it by combinatorial methods [cf. loc. cit., Theorem 4.1]. The goal of this note is to give a geometric interpretation of this formula for the Coxeter groups that are isomorphic to the Weyl group of a split group over a finite field. This interpretation rests on a result of the author [\textit{S. Morel}, J. Am. Math. Soc. 21, No. 1, 23-61 (2008; Zbl 1225.11073), Theorem 3.3.5] that expresses the intermediate extension of a pure perverse sheaf as a ``weight truncation'' of the usual direct image. Kazhdan-Lusztig polynomials; Coxeter groups; intersection complexes; Weyl groups Morel, S.: Note sur LES polynômes de Kazhdan-Lusztig. Math. Z. 268, 593-600 (2011) Hecke algebras and their representations, Reflection and Coxeter groups (group-theoretic aspects), Combinatorial aspects of representation theory, Modular and Shimura varieties, Étale and other Grothendieck topologies and (co)homologies Note on the Kazhdan-Lusztig polynomials. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main result of this paper is the following. Let \(R\) be a discrete valuation henselian ring with quotient field \(K\) and \({\mathfrak X}\to \text{Spec} (R)\) be a smooth projective curve over \(\text{Spec} (R)\). Denote by \(X={\mathfrak X} \otimes_RK\) the generic fiber. Consider an algebraic cover or \(G\)-cover \(Y\to X\) of \(X\) defined a priori over \(\overline K\). If the maximal ideal is ``not bad'' for the \((G)\)-cover \(Y\to X\), then the \((G)\)-cover has a model over \(M^{ur}\), where \(M\) is the field of moduli and \(M^{ur}\) denotes the maximal unramified algebraic extension of \(M\). Here the ``bad'' primes are those where the curve \(X\) has bad reduction, where two points of the branch locus meet, or which divide the order of the geometric monodromy group. This result is a generalisation of a theorem by \textit{P. Dèbes} and \textit{D. Harbater} [J. Reine Angew. Math. 498, 223-236 (1998; Zbl 0905.14015)] concerning only \(G\)-covers of \(\mathbb{P}^2\). The proof is based on the ``théorème de spécialisation'' of A. Grothendieck. covers of arithmetic surfaces; théorème de spécialisation of A. Grothendieck; algebraic cover; geometric monodromy group M. Emsalem, On reduction of covers of arithmetic surfaces, 1997 Ramification problems in algebraic geometry, Minimal model program (Mori theory, extremal rays), Arithmetic ground fields for curves, Special surfaces On reduction of covers of arithmetic surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book consists of eight chapters plus a five-page appendix on basic finite field theory. The first chapter is an overview largely concentrating on \textit{Multivariate Public Key Cryptosystems} (MPKC)s which have been viewed as possible alternatives to PKCs such as RSA.
Essentially an MPKC is a cryptosystem in which the public key is a finite set of multivariate polynomials \(f_j\in\mathbb{F}[x_1,\dots,x_n]\) where \(\mathbb{F}\) is a finite field. In this scenario, our usual cast of cryptographic characters, Alice and Bob, communicate as follows. If Alice wishes to send a message \((m_1,\dots,m_n)\in\mathbb{F}^n\) to Bob, she obtains Bob's public key \(f_j\) and computes \(c_j=f_j(m_1,\dots,m_n)\) for \(j=1,\dots,\ell\) and sends ciphertext \((c_1,\dots,c_\ell)\) to him. Bob's private key involves information about \(f_j\) which is a means of unlocking the one-way system \(f_j(m_1^\prime,\dots,m_n^\prime)=c_j\) for \(j=1,\dots,\ell\) without which it is computationally infeasible to solve it.
Reasons for the MPKC's role as a possible alternative to standard PKCs such as RSA include the fact that the former are much more computationally efficient than the latter, whose security is based upon the presumed difficulty of factoring large integers -- the \textit{Integer Factoring Problem} (IFP). Since the (general) problem of solving a set of multivariate polynomial equations over a finite field is (provably) an NP-hard problem, quantum computers could not be expected to efficiently find a solution set. Yet quantum computers would solve the IFP, making such PKCs as RSA obsolete, but not MPKCs.
Chapter two delves into \textit{Matsumoto-Imai Cryptosystems} (MIC)s, one of the first successful attempts to build an MPKC. The basic idea for this type of cryptosystem is to utilize a field extension \(K\) of degree \(n\) over a finite field \(\mathbb F\) (considered as an \(n\)-dimensional vector space over \(\mathbb{F}\)) and look for invertible maps on \(K\) which are then transformed into invertible maps over \(\mathbb {F}^n\). Attacks on MIC and its variants are considered and complexity issues discussed.
Chapter three is dedicated to the family of three signature schemes called \textit{Oil-Vinegar}, which may be seen as arising from attacks on MICs. Chapter four investigates a mechanism for making the notion of an MPKC more secure using what is called a \textit{Hidden Field Equation} (HFE) invented in 1996. Chapter five continues the investigation of extending MICs by looking at \textit{internal perturbations} which were motivated by a desire to resist certain attacks without sacrificing efficiency.
Chapter six has its origins in algebraic geometry since the \textit{triangular schemes} which it delineates, are motivated by the difficulty of decomposing a composition of invertible nonlinear polynomials maps -- a problem closely related to the well-known Jacobian conjecture. Chapter six is the longest and most involved section, which collates much of what has been previously discussed. Chapter seven continues the foray into algebraic geometry by considering \textit{direct attacks} on MPKCs. Typically, attacks employ numerical methods which are often based on ideas of Newton since it suffices to find approximate solutions to a set of equations. However, when the solutions of the equations are considered over a finite field, numerical methods are not applicable and exact solutions are required so algebraic geometry comes into play.
The concluding chapter eight looks to the future of MPKC research. This ten-page chapter summarizes and gathers results from the literature on the construction of MPKCs, their security, practical applications, and a brief look at underlying mechanisms.
Although the authors state, in their introduction, that this could be used as a textbook ``suitable for beginning graduate students in mathematics or computer science'', it is hampered by the fact that there are no exercises, no theorems or proofs, indeed no rigorous mathematical development. Thus, especially given the very specialized topic matter, it would not be suitable for students in mathematics at any level. The authors admit, in the introduction, that ``this book has been written from the computational perspective'', and this is evident. As a textbook, however, even in computer science, it might be suitable as a reference for specific aspects of an advanced course in cryptology with MPKCs as one of the topics. Certainly anyone interested in this area of cryptology would benefit from having this book as part of their library. polynomials; multivariate; cryptology; public-key Ding, J., Gower, J.E., Schmidt, D.S.: Multivariate public key cryptosystems, advances in information security, 25. Springer, Berlin, Heidelberg (2006) Cryptography, Research exposition (monographs, survey articles) pertaining to information and communication theory, Authentication, digital signatures and secret sharing, Applications to coding theory and cryptography of arithmetic geometry, Algebraic coding theory; cryptography (number-theoretic aspects) Multivariate public key cryptosystems | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{St. Lichtenbaum} [Ann. Math. (2) 170, No. 2, 657--683 (2009; Zbl 1278.14029)] conjectured that for an arithmetic scheme there is a Grothendieck topology such that the Euler characteristic for the cohomology with constant sheaf of the integers can give the leading term of the Dedekind zeta function at zero.
In this paper, the author discusses the case of the ring of algebraic integers of a number field. The corresponding topos in the conjecture is studied under the topological assumptions. For such a topos, the expected properties are given and several related results on cohomology groups and fundamental groups are also obtained. Then the author proves the main result of the Lichtenbaum's formalism on the Euler characteristic and the leading term of the zeta function at zero. Grothendieck topology; topos; étale cohomology; fundamental group; Dedekind zeta function; Euler characteristic Morin, B.: The Weil-étale fundamental group of a number field I. Kyushu J. Math. 65 (2011, to appear) Étale and other Grothendieck topologies and (co)homologies, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) The Weil-étale fundamental group of a number field. I | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Grothendieck-Katz \(p\)-curvature conjecture is an analogue of the Hasse principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its \(p\)-curvature vanishes modulo \(p\), for almost all primes \(p\). We prove that if the variety is a generic curve, then every simple closed loop on the curve has finite monodromy. Grothendieck-Katz \(p\)-curvature conjecture Curves of arbitrary genus or genus \(\ne 1\) over global fields, Rigid analytic geometry, Fibrations, degenerations in algebraic geometry The \(p\)-curvature conjecture and monodromy around simple closed loops | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this paper is to prove the topological mirror symmetry conjecture of \textit{T. Hausel} and \textit{M. Thaddeus} [C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 4, 313--318 (2001; Zbl 1009.14005); Invent. Math. 153, No. 1, 197--229 (2003; Zbl 1043.14011)] for the moduli space of strongly parabolic Higgs bundles of rank two and three, with full flags, for any generic weights. Although the main theorem is proved only for rank at most three, most of the results are proved for any prime rank.
This paper is organized as follows: Section 1 is an introduction to the subject. Section 2 reviews basic facts about parabolic Higgs bundles and their moduli. The authors also recall how in the parabolic setting moduli spaces for different degrees $d$ are isomorphic (with a change inparabolic weights). In Section 3 they recall the SYZ mirror symmetry result for the parabolic Hitchin system, review the stringy $E$-polynomials, describe the topological mirror symmetry conjecture of Hausel-Thaddeus, and state and prove their result. Section 4 is devoted to the calculation of the contribution to the variant part of the $E$-polynomial of the $\mathrm{SL}(n, \mathbb{C})$-modulispace arising only from the \(C^\ast\)-fixed point loci of type \((1,1,\dots,1)\). In Section 5 the authors recall some classical results on Prym varieties of unramified covers. These are used in Section 6, where the contribution from the fixed point loci of non-trivial elements of the group $\Gamma_n$ (of $n$-torsion points of $\mathrm{Pic}^0(X)$ with $X$ a Riemann surface) to the stringy E-polynomial of $\mathrm{PGL}(n, \mathbb{C})$-moduli space is calculated. Higgs bundles; parabolic structures; mirror symmetry; E-polynomials Vector bundles on curves and their moduli, Jacobians, Prym varieties, Relationships between algebraic curves and integrable systems Topological mirror symmetry for parabolic Higgs bundles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we develop a certain combinatorial version of the theory of Belyi cuspidalization developed by Mochizuki. Write \({\overline{\mathbb{Q}}} \subseteq \mathbb{C}\) for the subfield of algebraic numbers \(\in \mathbb{C} \). We then apply this theory of combinatorial Belyi cuspidalization to certain natural closed subgroups of the Grothendieck-Teichmüller group associated to the field of \(p\)-adic numbers [where \(p\) is a prime number] and to stably \(\times \mu\)-indivisible subfields of \({\overline{\mathbb{Q}}} \), i.e., subfields for which every finite field extension satisfies the property that every nonzero divisible element in the field extension is a root of unity. anabelian geometry; Belyi cuspidalization; Grothendieck-Teichmüller group Coverings of curves, fundamental group Combinatorial Belyi cuspidalization and arithmetic subquotients of the Grothendieck-Teichmüller group | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the present article, we establish new evidence in support of the Strong Factorial Conjecture of \textit{E. Edo} and \textit{A. van den Essen} [J. Algebra 397, 443--456 (2014; Zbl 1405.13038)] by proving it in several special cases. For example, we show that the conjecture holds for powers of linear forms and sums of prime powers of the variables. Finally, we show one way of constructing new examples of polynomials satisfying the conjecture using known examples. polynomial automorphisms; Jacobian conjecture; factorial conjecture; rigidity conjecture; irreducible polynomials Polynomial rings and ideals; rings of integer-valued polynomials, Jacobian problem On the incompatibility of Diophantine equations arising from the strong factorial conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simple complex algebraic group, and let \(K \subset G\) be a reductive subgroup such that the coordinate ring of \(G/K\) is a multiplicity-free \(G\)-module. We consider the \(G\)-algebra structure of \(\mathbb{C}[G/K]\) and study the decomposition into irreducible summands of the product of irreducible \(G\)-submodules in \(\mathbb{C}[G/K]\). When the spherical roots of \(G/K\) generate a root system of type \(\mathsf{A}\), we propose a conjectural decomposition rule, which relies on a conjecture of Stanley on the multiplication of Jack symmetric functions. With the exception of one case, we show that the rule holds true whenever the root system generated by the spherical roots of \(G/K\) is a direct sum of subsystems of rank 1. reductive spherical pairs; multiplicity-free actions; coordinate rings of spherical varieties; Jack polynomials Compactifications; symmetric and spherical varieties, Combinatorial aspects of representation theory, Representation theory for linear algebraic groups On the multiplication of spherical functions of reductive spherical pairs of type A | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper gives a short but comprehensive survey of problems and results concerning mappings induced on certain fields (or rings) by polynomials. For a given field \(K\) the following property is considered:
(P) There is no infinite subset \(E\) in \(K\) for which there exists a nonlinear polynomial \(P\) defined on \(K\) such that \(P(E) = E\).
This property has been first investigated by the author [Acta Arith. 7, 241-249 (1962; Zbl 0125.00901)] who proved that any finite extension of the rationals has property (P). Several extensions of property (P) have been considered by K. K. Kubota, D. J. Lewis, the author and the reviewer. Related questions concerning cycles for polynomials are also given. Among the various problems which are proposed here, the following ones are widely open:
Problem 1-2: characterize fields which have property (P).
Problem 6: improve the estimate for the maximum length of cycles (lying in a given algebraic number field \(K)\) of a nonlinear polynomial \(P\).
In the case where \(K = \mathbb{Q}\) and \(P\) is monic with rational integral coefficients, the author proves that the length of cycles does not exceed 2. extensions of fields; algebraic numbers; polynomial mappings; survey; polynomials; cycles; length of cycles Polynomials in general fields (irreducibility, etc.), Rational and birational maps Polynomial mappings. (A survey of results and problems) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(T=\mathbb{G}_{m}^{n}\) be an \(n\)-dimensional algebraic torus over \(\overline{\mathbb{Q}}\), where \(n\geq 2\). For \(0\leq r\leq n-1\), let \(\mathcal{H}_{\,r}\) denote the union of all \(r\)-dimensional algebraic subgroups of \(T\). Set \(\mathcal{H}=\bigcup_{\,r=0}^{\,n-1}\mathcal{H}_{\,r}\).
In [[1]: \textit{E. Bombieri, D. Masser} and \textit{U. Zannier}, ``Intersecting a curve with algebraic subgroups of multiplicative groups'', Int. Math. Res. Not. 1999, No. 20, 1119--1140 (1999; Zbl 0938.11031)], the authors studied the intersection of a given closed and irreducible curve \(C\subset T\) with \(\mathcal{H}\). In particular, they showed that if \(C\) is not contained in a translate of a proper subtorus of \(T\), then the set \(C\cap\mathcal{H}(\overline{\mathbb{Q}})\) has bounded (Weil) height. The paper under review is concerned with a generalization of this result to higher-dimensional subvarieties \(X\subset T\). To explain the sort of results actually proved in this paper, we go back to the works [[2]: \textit{E. Bombieri} and \textit{U. Zannier}, ``Algebraic points on subvarieties of \(\mathbb{G}_{m}^{n}\)'', Int. Math. Res. Not. 1995, No. 7, 333--347 (1995; Zbl 0848.11030)] and [[3]: \textit{U. Zannier}, ``Appendix'', in ``Polynomials with special regard to reducibility. With an Appendix by Umberto Zannier'' by A. Schinzel. Encyclopedia of Mathematics and its Applications 77. Cambridge University Press (2000; Zbl 0956.12001), pp. 517--539] by the first and third authors.
Let \(X^{\circ}\) denote the complement in \(X\) of the union of all subvarieties of \(X\) which are translates of nontrivial subtori of \(T\). Then \(X^{\circ}\) is a Zariski-open subset of \(X\) [2]. Further, if \(X\) is defined over \(\overline{\mathbb{Q}}\), then \(X^{\circ}\cap\mathcal H_{1}\) is a set of bounded height [3]. In the paper under review the authors introduce a new set \(X^{{\circ}a}\), analogous to \(X^{\circ}\) and contained in it if \(X\neq T\), and show that it is (Zariski) open in \(X\) (in fact, a sharper ``structure theorem'' for \(X^{{\circ}a}\) is obtained. See Theorem 1.4 of the paper). The set \(X^{{\circ}a}\) is defined as the complement in \(X\) of the union of all ``anomalous'' subvarieties of \(X\). A positive-dimensional irreducible subvariety \(Y\) of \(X\) is \textit{anomalous} if it lies in a translate \(K\) of an algebraic subgroup of \(T\) and its dimension is strictly larger than \(\text{dim}\, X+\text{dim}\,K-n\) (so \(X\) and \(K\) do \textit{not} meet properly since \(\text{dim}\,(X\cap K)\geq \text{dim}\, Y>\text{dim}\,X+\text{dim}\,K-n\)).
The authors also state the following \textit{Bounded Height Conjecture}. Let \(X\) be an irreducible subvariety of \(T\) of dimension \(r\). Then \(X^{{\circ}a}\cap\mathcal H_{\,n-r}\) is a set of bounded height. When \(X\) is a curve \(C\) not contained in a translate of dimension \(n-1\), then \(X^{{\circ}a}=C\) and the conjecture is true by the result on curves quoted above. On the other hand, if \(r=n-1\) (i.e., if \(X\) is a hypersurface in \(T\)) then \(X^{{\circ}a}=X^{\circ}\) and the conjecture is true by the result from [3] cited above. No other instances where the conjecture is true are known, but the authors have promised to settle the case of planes in \(T\) in a subsequent publication.
The second main result of the paper establishes the existence of a finite collection \(\Psi\) of translations \(S\) of tori by torsion points, satisfying \(\text{dim}(X\cap S)\geq \text{dim}\,S-1\), such that \(X\cap\mathcal H_{\,1}=\bigcup_{S\in\Psi}(X\cap S)\cap \mathcal H_{\,1}\). The significance of this result is that it reduces the problem of describing \(X\cap\mathcal H_{\,1}\) for general \(X\) to the hypesurface case since \(X\cap S\) may be regarded as a hypersurface in \(S\) and \(S\) is essentially \(\mathbb G_{m}^{d}\) for some \(d\). No analogous description of \(X\cap\,\mathcal H_{\,2}\) is known at present. The paper also contains a result (Theorem 1.6) on lacunary polynomials with algebraic coefficients which has implications for irreducibility questions. This result (which, in the interest of brevity, we do not state here) extends work of Schinzel and of the first and third authors in [3].
The paper also discusses a set \(X^{ta}\) which is obtained by removing from \(X\) all ``torsion-anomalous'' subvarieties of \(X\) (to define a torsion-anomalous subvariety, simply repeat the definition of ``anomalous'' above specializing \(K\) to a translate of the form \(gH\), where \(g\) is a torsion element of \(T\) and \(H\) is an algebraic subgroup of \(T\)). The following conjectures are discussed: (a) Let \(X\) be an irreducible subvariety of \(T\) defined over \(\mathbb C\). Then \(X^{ta}\) is Zariski-open in \(X\), and (b) if \(X\) (as in (a)) has dimension \(r\), then \(X^{ta}\cap\mathcal H_{n-r-1}\) is a finite set. The authors also discuss generalizations of the above conjectures to the case of semi-abelian varieties. Finally, the third author corrects an inaccuracy which appears in the proof of Theorem 2 in [3]. heights; tori; lacunary polynomials Bombieri, E., Masser, D., Zannier, U.: Anomalous subvarieties--structure theorems and applications. Int. Math. Res. Not. \textbf{2007}: Article ID rnm057 (2007) Heights, Arithmetic varieties and schemes; Arakelov theory; heights Anomalous subvarieties -- structure theorems and applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let S be a complex projective nonsingular minimal surface of general type and let \({\mathcal M}(S)\) be the coarse moduli space of complex structures on the oriented topological 4-manifold underlying S. It is known that \({\mathcal M}(S)\) is a quasi-projective variety. The paper under review is the third of a series [(1) J. Differ. Geom. 19, 483-515 (1984; Zbl 0549.14012); (2) Algebraic Geometry, Open Problems, Proc. Conf., Ravello/Italy 1982, Lect. Notes Math. 997, 90-112 (1983; Zbl 0517.14011); (4) J. Differ. Geom. 24, 395-399 (1986)] devoted to the study of general properties of \({\mathcal M}(S)\). This study was carried out by using a test class of simply connected surfaces obtained by deforming bidouble covers of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\), i.e. Galois covers with group \(({\mathbb{Z}}/2)^ 2\). The interest in these bidouble covers is evident in view of the analogy with hyperelliptic curves in dimension \(1\).
Here the author studies in the large the deformations of these surfaces. A bidouble cover as above looks like the subvariety of the total space of the bundle \({\mathcal O}_{{\mathbb{P}}^ 1\times {\mathbb{P}}^ 1}(a,b)\oplus {\mathcal O}_{{\mathbb{P}}^ 1\times {\mathbb{P}}^ 1}(n,m)\) defined by equations \(z^ 2=f(x,y)\), \(w^ 2=g(x,y)\), where f and g are bihomogeneous forms of bidegrees (2a,2b) and (2n,2m) respectively. Let \({\mathcal N}_{(a,b),(n,m)}\) be the subset of the moduli space corresponding to smooth natural deformations; the author supplies a complete description of the closure of \({\mathcal N}_{(a,b),(n,m)}\), for \(a>2n\), \(m>2b\). This is achieved by exploiting the relations between deformations of bidouble covers of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) and degenerations of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) to normal surfaces with certain singularities, which the author calls ''1/2 rational double points'' and studies in great detail. minimal surface of general type; coarse moduli space; bidouble covers; deformations; 1/2 rational double points Catanese F.: Automorphisms of rational double points and moduli spaces of surfaces of general type. Compos. Math. 61(1), 81--102 (1987) Families, moduli, classification: algebraic theory, Coverings in algebraic geometry, Formal methods and deformations in algebraic geometry, Special surfaces, Singularities of surfaces or higher-dimensional varieties Automorphisms of rational double points and moduli spaces of surfaces of general type | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials {Let \(Fl_n\) be the manifold of complete flags in the \(n\)-dimensional vector space \(\mathbb C^n\). Inspired from ideas from string theory, recently the concept of quantum cohomology ring \(QH^*(X,\mathbb Z)\) of a Kähler algebraic manifold \(X\) has been defined. Then
\[
QH^*(Fl_n,\mathbb Z)\cong H^*(Fl_n,\mathbb Z) \otimes\mathbb Z[q_1,\dots,q_{n-1}],
\]
where \(H^*(X,\mathbb Z)\) is the usual cohomology ring of \(Fl_n\) and \(q_1,\dots,q_{n-1}\) are formal variables (deformation parameters). So, the additive structures of the two cohomology rings are essentially the same. The multiplicative structure of \(H^*(X,\mathbb Z)\) can be recuperated from the multiplicative structure of \(QH^*(X,\mathbb Z)\) by taking \(q_1=\cdots=q_{n-1}=0\). The structure constants for the quantum cohomology are the 3-point Gromov-Witten invariants of genus zero. Recently, Givental, Kim and Ciocan-Lafontaine found a canonical isomorphism
\[
QH^*(X,\mathbb Z)\cong\mathbb Z[q_1,\dots,q_{n-1}][x_1,\dots,x_n]/I_n^q,
\]
where \(x_1,\dots,x_n\) are variables and \(I_n^q\) is a certain ideal which can be explicitly described. This isomorphism extends an old isomorphism of Borel for the ordinary cohomology ring. The next problem naturally arising in the theory of quantum cohomology of the flag manifolds is to find an algebraic/combinatorial method for computing the structure constants of quantum multiplication in the basis of Schubert classes (the Gromov-Witten invariants). The aim of the paper under review is to solve this problem completely.} flag varieties; Schubert varieties; quantum cohomology ring; complete flags; Gromov-Witten invariants S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials , J. Amer. Math. Soc., 168 (1997), 565--596. JSTOR: Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Quantum Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors give a description of the intersection cohomology groups of the Schubert varieties in partial flag manifolds over symmetrizable Kac-Moody Lie algebras in terms of parabolic Kazhdan-Lusztig polynomials introduced by \textit{V. Deodhar} [J. Algebra 111, 483-506 (1987; Zbl 0656.22007)]. Grassmanians; Schubert varieties; flag manifolds; intersection cohomology groups; symmetrizable Kac-Moody Lie algebras Kashiwara, M., \& Tanisaki, T. (2002). Parabolic Kazhdan-Lusztig polynomials and Schubert varieties. J. Algebra, 249, 306--325. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Parabolic Kazhdan-Lusztig polynomials and Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(S\) be a fixed Noetherian scheme and \(\mathcal S\) the category of separated, essentially finite-type, finite tor-dimension schemes \(x : X \rightarrow S\) over \(S\). For any such scheme \(X\), let \(\text{D}_{\text{qc}}(X)\) denote the derived category of the category of complexes of \({\mathcal O}_X\)-modules with quasi-coherent cohomology sheaves. For \(A\), \(B\) in \(\text{D}_{\text{qc}}(X)\), consider the graded abelian group:
\[
\text{E}_X(A,B) := { \bigoplus_{i \in {\mathbb Z}}} \text{Hom}_{\text{D}(X)}(A,B[i])\, .
\]
For \(A\), \(B\), \(C\) in \(\text{D}_{\text{qc}}(X)\) there exists an obvious graded bilinear composition map:
\[
\text{E}_X(B,C)\times \text{E}_X(A,B) \longrightarrow \text{E}_X(A,C)\, .
\]
Via this composition map, \(H_X := \text{E}_X({\mathcal O}_X,{\mathcal O}_X) \simeq \bigoplus_{i \geq 0}\text{H}^i(X,{\mathcal O}_X)\) becomes a commutative-graded ring and \(\text{E}_X(A,B)\) a symmetric (left and right) graded \(H_X\)-module. Putting \(H := H_S\), the category \(\text{D}_X\) whose objects are the objects of \(\text{D}_{\text{qc}}(X)\) and whose Hom-groups are \(\text{E}_X(A,B)\) is a \(H\)-\textit{graded category}. If \(f : X \rightarrow Y\) is a morphism in \(\mathcal S\) then the pseudofunctors \(\text{R}f_\ast : \text{D}_{\text{qc}}(X) \rightarrow \text{D}_{\text{qc}}(Y)\) and \(\text{L}f^\ast : \text{D}_{\text{qc}}(Y) \rightarrow \text{D}_{\text{qc}}(X)\) induce pseudofunctors of \(H\)-graded categories \(\text{R}f_\ast : \text{D}_X \rightarrow \text{D}_Y\) and \(\text{L}f^\ast : \text{D}_Y \rightarrow \text{D}_X\).
\textit{S. Nayak} [Adv. Math. 222, No. 2, 527--546 (2009; Zbl 1175.14003)], unifying the local and global Grothendieck duality theories, constructed, under the above hypotheses, a \textit{twisted inverse image} pseudofunctor \(f_+^! : \text{D}^+_{\text{qc}}(Y) \rightarrow \text{D}^+_{\text{qc}}(X)\) which is pseudofunctorially right-adjoint to \(\text{R}f_\ast : \text{D}^+_{\text{qc}}(X) \rightarrow \text{D}^+_{\text{qc}}(Y)\) if \(f\) is proper, and such that \(f_+^! = f^\ast\) if \(f\) is essentially étale (which means that \({\mathcal O}_{X,x}\) is formally étale over \({\mathcal O}_{Y,f(x)}\), \(\forall \, x \in X\)). One can show that \(f_+^!\) can be extended to a pseudofunctor of \(H\)-graded categories \(f^! : \text{D}_Y \rightarrow \text{D}_X\) such that \(f^!C = f_+^!{\mathcal O}_Y\otimes_{{\mathcal O}_X}^{\text{L}}\text{L}f^\ast C\) for \(C \in \text{D}_{\text{qc}}(Y)\).
Finally, for any object \(X \rightarrow S\) of \(\mathcal S\), consider the \textit{pre-Hochschild complex} \({\mathcal H}_X := \text{L}\delta_X^\ast \text{R}\delta_{X\ast}{\mathcal O}_X\), where \(\delta_X : X \rightarrow X\times_SX\) is the diagonal morphism. Using the properties of these complexes (which can be found in the paper of \textit{R.-O. Buchweitz} and \textit{H. Flenner} [Adv. Math. 217, No. 1, 205--242 (2008; Zbl 1140.14015)]) and of the twisted inverse image pseudofunctor \(f^!\), the authors of the paper under review develop a \textit{bivariant theory} (in the sense of \textit{W. Fulton} and \textit{R. MacPherson} [``Categorical framework for the study of singular spaces'', Mem. Am. Math. Soc. 243, 165 p. (1981; Zbl 0467.55005)]) on the category \(\mathcal S\), with values in the \(H\)-graded categories \(\text{D}_X\), \(X \in {\mathcal S}\), for the pseudofunctors \((-)^\ast\), \((-)_\ast\) and \((-)^!\), with proper morphisms as \textit{confined maps}, and with cartesian squares with flat bottom as \textit{independent squares}. This theory associates to a morphism \(f : (X \overset{x}\rightarrow S) \rightarrow (Y \overset{y}\rightarrow S)\) the graded \(H\)-module:
\[
\text{HH}^\ast(f) := \text{E}_X({\mathcal H}_X,f^!{\mathcal H}_Y) = { \bigoplus_{i\in {\mathbb Z}}}\text{Hom}_{\text{D}(X)}({\mathcal H}_X, f^!{\mathcal H}_Y[i])
\]
so that the associated cohomology groups are:
\[
\text{HH}^i(X/S) := \text{HH}^i(\text{id}_X) = \text{Ext}^i_{{\mathcal O}_X}({\mathcal H}_X,{\mathcal H}_X)
\]
and the associated homology groups are:
\[
\text{HH}_i(X/S) := \text{HH}^{-i}(x) = \text{Ext}^{-i}_{{\mathcal O}_X}({\mathcal H}_X,x^!{\mathcal O}_S)\, .
\]
Most of the proofs consist of the (non-trivial) verification of the commutativity of certain diagrams.
These results lay the foundation for the construction, in a sequel of this paper, of the \textit{fundamental class} of a \textit{flat} \(f\) as above, which is a natural functorial map:
\[
\text{c}_f : \text{L}\delta^\ast_X\text{R}\delta_{X\ast}\text{L}f^\ast \longrightarrow f^!\text{L}\delta^\ast_Y\text{R}\delta_{Y\ast}
\]
satisfying a transitivity relation with respect to the composition of morphisms in \(\mathcal S\). Hochschild homology; bivariant theory; Grothendieck duality; fundamental class Tarrío, L. Alonso; Lopez, A. Jeremías; Lipman, J.; Bivariance: Grothendieck duality and Hochschild homology I: Construction of a bivariant theory, Asian J. Math. 15, 451-498 (2011) (Co)homology theory in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Bivariance, Grothendieck duality and Hochschild homology. I: Construction of a bivariant theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Each cohomology ring of a Grassmannian or flag variety has a basis of Schubert classes indexed by the elements of the corresponding Weyl group. Classical Schubert calculus computes the cohomology rings of Grassmannians and flag varieties in terms of the Schubert classes. This paper is ``doing Schubert calculus'' in the equivariant cohomology rings of Peterson varieties.
The Peterson variety is a subvariety of the flag variety \(G/B\) parameterized by a linear subspace \(H_{\mathrm{Pet}} \subseteq \mathfrak g\) and a regular nilpotent operator \(N_0 \in \mathfrak g\). We can define the Peterson variety as
\[
\mathrm{Pet}=\{gB\in G\backslash B:\mathrm{Ad}(g^{-1})N_0 \in H_{\mathrm{Pet}}\}.
\]
Peterson varieties were introduced by Peterson in the 1990s. Peterson constructed the small quantum cohomology of partial flag varieties from what are now Peterson varieties. \textit{B. Kostant} [Sel. Math., New Ser. 2, No. 1, 43--91 (1996; Zbl 0868.14024)] used Peterson varieties to describe the quantum cohomology of the flag manifold and \textit{K. Rietsch} [Nagoya Math. J. 183, 105--142 (2006; Zbl 1111.14048)] gave the totally non-negative part of type A Peterson varieties. \textit{E. Insko} and \textit{A. Yong} [Transform. Groups 17, No. 4, 1011--1036 (2012; Zbl 1267.14066)] explicitly identified the singular locus of type A Peterson varieties and intersected them with Schubert varieties. \textit{M. Harada} and \textit{J. Tymoczko} [Proc. Lond. Math. Soc. (3) 103, No. 1, 40--72 (2011; Zbl 1219.14065)] proved that there is a circle action \(\mathbb S^1\) which preserves Peterson varieties. In this paper the authors study the equivariant cohomology of the Peterson variety with respect to this action and also they use GKM theory as a model for studying equivariant cohomology, but Peterson varieties are not GKM spaces under the action of \(\mathbb S^1\). Using work by Harada and Tymoczko [Zbl 1219.14065] and \textit{M. Precup} [Sel. Math., New Ser. 19, No. 4, 903--922 (2013; Zbl 1292.14032)], they construct a basis for the \(\mathbb S^1\)-equivariant cohomology of Peterson varieties in all Lie types. This construction gives a set of classes which we call Peterson Schubert classes. The name indicates that the classes are projections of Schubert classes, they do not satisfy all the classical properties of Schubert classes.
Classical Schubert calculus asks how to multiply Schubert classes; here, the authors asks how to multiply in the basis of Peterson Schubert classes. She gives a Monk's formula for multiplying a ring generator and a module generator, and a Giambelli's formula for expressing any Peterson Schubert class in the basis in terms of the ring generators. Peterson variety; equivariant cohomology; Monk's rule; Giambelli's formula; Schubert calculus Drellich, Elizabeth, Monk's rule and Giambelli's formula for Peterson varieties of all Lie types, J. Algebraic Combin., 41, 2, 539-575, (2015) Classical problems, Schubert calculus, Equivariant homology and cohomology in algebraic topology, Grassmannians, Schubert varieties, flag manifolds Monk's rule and Giambelli's formula for Peterson varieties of all Lie types | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is the first of a series of papers which review the geometric construction of the double affine Hecke algebra via affine flag manifolds and explain the main results of the authors on its representation theory. There are also some simplifications of the original arguments and proofs for some well-known results for which there exists no reference.
This paper concerns the most basic facts of the theory: the geometric construction of the double affine Hecke algebra via the equivariant, algebraic K-theory and the classification of the simple modules of the category \(\mathcal O\) of the double affine Hecke algebra. It is our hope that by providing a detailed explanation of some of the difficult aspects of the foundations, this theory will be better understood by a wider audience.
This paper contains three chapters. The first one is a reminder on \(\mathcal O\)-modules over non Noetherian schemes and over ind-schemes. The second one deals with affine flag manifolds. The last chapter concerns the classification of simple modules in the category \(\mathcal O\) of the double affine Hecke algebra. double affine Hecke algebras; degenerate Hecke algebras; simple modules in category \(\mathcal O\); simple spherical representations; induced modules; rational Cherednik algebras; spherical finite-dimensional modules M. Varagnolo and E. Vasserot, Double affine Hecke algebras and affine flag manifolds. I, Affine flag manifolds and principal bundles, 233--289, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2010. Hecke algebras and their representations, Grassmannians, Schubert varieties, flag manifolds, Grothendieck groups, \(K\)-theory, etc., Quantum groups (quantized enveloping algebras) and related deformations Double affine Hecke algebras and affine flag manifolds. I. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce the notion of a cominuscule point in a Schubert variety in a generalized flag variety for a semisimple group. We derive formulas expressing the Hilbert series and multiplicity of a Schubert variety at a cominuscule point in terms of the restrictions of classes in torus-equivariant \(K\)-theory and cohomology to that point, generalizing previously known formulas for flag varieties of cominuscule type. Thus, we can calculate Hilbert series and multiplicities in cases where these were previously unknown. The formulas for Schubert varieties are special cases of more general formulas valid at generalized cominuscule points of schemes with torus actions. flag variety; Schubert variety; cominuscule; minuscule; equivariant; \(K\)-theory Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Equivariant \(K\)-theory Cominuscule points and Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let $\mathbf{k}$ be a field of characteristic zero, $m$ be an integer with $3\le m$, $S$ be the polynomial ring $\mathbf{k}[x_{i,j}\mid 1\le i\le j\le m]$, and $\mathcal S$ be the $m\times m$ symmetric matrix with $x_{i,j}$ in row $i$ and column $j$ for $1\le i\le j\le m$. This paper is concerned with two specializations of $\mathcal S$. In one specialization, $x_{m-1,m-1}$ is set equal to $x_{m,m}$. In the other specialization, the bottom right hand corner of $\mathcal S$ is set equal to zero. Let $R$ be the polynomial ring which is the image of $S$ under the specialization. \par For each specialization various algebraic and geometric objects are considered. Let $f$ be the determinant of $\mathcal S$ after specialization, $h(f)$ be the Hessian matrix of second order partial derivatives of $f$, $J$ be the ideal generated by the partial derivatives of $f$, and $P$ be the ideal generated by the $(m-1)\times (m-1)$ minors of the specialization. The ``polar map'' defined by the partial derivatives of $f$, the image, $V(f)$, of the polar map, and the dual variety, $V(f)^*$, of $V(f)$ are also studied. \par Some of the questions that are answered include the following. What is the codimension of $J$? Is $J$ contained in $P$? What are the properties of the hypersurface ring $R/(f)$? Is it a domain? Is it normal? Does the determinant of $h(f)$ vanish? How many linear relations are there on the generators of $J$? Is the polar map birational? What are the properties of $R/P$? Is it Cohen-Macaulay? Is it a domain? Is it normal? What is its codimension? What properties does that rational map of projective spaces which is defined by the generators of $P$ have? Is it birational? What are the defining equations of its image? What is the homogeneous coordinate ring of the polar variety? What is the analytic spread of $J$? What is the dimension of $V(f)^*$? Is $V(f)^*$ arithmetically Cohen-Macaulay? When is $V(f)^*$ arithmetically Gorenstein? \par The analogous program for generic matrices, rather than generic symmetric matrices, is carried out in [\textit{R. Cunha} et al., Int. J. Algebra Comput. 28, No. 7, 1255--1297 (2018; Zbl 1403.13020)]. generic symmetric matrix; homaloidal polynomials; polar map; gradient ideal; linear syzygies; dual variety Linkage, complete intersections and determinantal ideals, Syzygies, resolutions, complexes and commutative rings, Rational and birational maps, Determinantal varieties, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Symmetry preserving degenerations of the generic symmetric matrix | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The jeu-de-taquin-based Littlewood-Richardson rule of \textit{H. Thomas} and \textit{A. Yong} [Algebra Number Theory 3, No. 2, 121--148 (2009; Zbl 1229.05285)] for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, \textit{A. Skovsted Buch} and \textit{M. J. Samuel} [J. Reine Angew. Math. 719, 133--171 (2016; Zbl 1431.19001)] developed a combinatorial theory of `unique rectification targets' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to \(K\)-theory. Separately, \textit{P.-E. Chaput} and \textit{N. Perrin} [J. Lie Theory 22, No. 1, 17--80 (2012; Zbl 1244.14036)] used the combinatorics of R. Proctor's `\(d\)-complete posets' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of \(d\)-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain \(K\)-theoretic Schubert structure constants in the Kac-Moody setting. unique rectification target; jeu de taquin; \(d\)-complete poset; Schubert calculus; Kac-Moody group Rahul Ilango, Oliver Pechenik, Michael Zlatin, Unique rectification in \(d\)-complete posets: towards the \(K\)-theory of Kac-Moody flag varieties, preprint 2018, 34 pages, arXiv:1805.02287. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Combinatorics of partially ordered sets, Grassmannians, Schubert varieties, flag manifolds Unique rectification in \(d\)-complete posets: towards the \(K\)-theory of Kac-Moody flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The existence of Ulrich bundles on projective varieties is relevant in the description of their geometric structure. Ulrich bundles are defined as vector bundles which are arithmetically Cohen-Macaulay, with maximal number of global sections in their first twist. It is conjectured that every smooth projective variety supports some Ulrich bundle \(E\), and indeed some large classes of projective varieties are known to satisfy the conjecture. On the other hand, it is hard in general to determine the minimal rank of Ulrich bundles, even for some specific natural classes of varieties \(X\). The athors consider the case where \(X\) is a double cover of \(\mathbb P^2\) branched along a curve \(B\) of even degree \(\geq 6\). A result of \textit{A. J. Parameswaran} and \textit{P. Narayanan} [J. Algebra 583, 187--208 (2021; Zbl 1473.14087)] proves that when \(B\) is general, then \(X\) has no Ulrich line bundles. The authors prove that for general \(B\) the double cover \(X\) supports Ulrich bundles of rank \(2\). The result is obtained by constructing on \(X\) special configurations of points, and then using the correspondence between finite sets with the Cayley-Bacharach property and rank \(2\) vector bundles on surfaces. Ulrich bundles; double planes Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Coverings in algebraic geometry Rank 2 Ulrich bundles on general double plane covers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Given positive integers \(m\) and \(p\) the map that associates homogeneous polynomials \(f_1,\dots ,f_m\) of degree \(m+p-1\) to its Wronskian defines a finite map between Grassmannian varieties
\[
Wr: Gr(m, \mathbb{C}_{m+p-1}[t])\to Gr(1,\mathbb{C}_{mp}[t])=\mathbb{P}(\mathbb{C}_{mp}[t]).
\]
Eisenbud and Harris have computed explicitly the degree \(d_{m,p}\) of this map in the work [\textit{D. Eisenbud} and \textit{J. Harris}, Invent. Math. 74, 371--418 (1983; Zbl 0527.14022)]. When one changes the base field from \(\mathbb{C}\) to \(\mathbb{R}\) the problem of computing the degree \(d_{m,p}'\) of \(Wr\) is known as inverse Wronski problem.
The authors show that \(d_{m,p}\) and \(d_{m,p}'\) are congruent modulo four. They do this using a general framework for congruences modulo four in Schubert calculus which they develop in the present paper. Schubert; Wronskian; Grassmanian \textsc{N.~Hein, F.~Sottile, and I.~Zelenko}, \textit{A congruence modulo four in real Schubert calculus}, J. Reine Angew. Math., 714 (2016), pp.~151-174. Classical problems, Schubert calculus, Semialgebraic sets and related spaces A congruence modulo four in real Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. \textit{F. Liu} and \textit{B. Osserman} [J. Algebr. Comb. 23, No. 2, 125--136 (2006; Zbl 1090.14009)] studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by \textit{C. Haase} and \textit{T. B. McAllister} [Contemp. Math. 452, 115--122 (2008; Zbl 1163.52006)], for this class of polytopes. The key ingredient in the proofs is the nearest neighbor interchange (NNI) move on graphs and a naturally arising piecewise unimodular transformation. We provide a generalization of the context in which the NNI moves appear, to connected graphs with the same degree sequence. We also show that, up to a dilation factor of 4 and an integer translation, all of these Liu-Osserman polytopes are reflexive. nearest neighbour interchange; Ehrhart polynomials; polytopes; cubic graphs; scissors congruence Enumeration in graph theory, Graph operations (line graphs, products, etc.), Dissections and valuations (Hilbert's third problem, etc.), Vector bundles on curves and their moduli Cubic graphs, their Ehrhart quasi-polynomials, and a scissors congruence phenomenon | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Tropical Newton-Puiseux polynomials, defined as piece-wise linear functions with rational coefficients of the variables, play a role as tropical algebraic functions. We provide explicit formulas for tropical Newton-Puiseux polynomials being the tropical zeroes of a univariate tropical polynomial with parametric coefficients.
For Part I, see [\textit{D. Grigoriev}, Lect. Notes Comput. Sci. 11077, 177--186 (2018; Zbl 1453.14148)]. tropical Newton-Puiseux polynomial; zeroes of tropical parametric polynomials Foundations of tropical geometry and relations with algebra Tropical Newton-Puiseux polynomials. II | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \((X,H)\) be a smooth \(K3\) surface \(X\) together with a nef divisor describing a birational contraction \(\pi: X \to Y\). Suppose that \(\pi\) resolves rational double points on \(Y\). By results of \textit{T. Bridgeland} [Invent. Math. 147, No. 3, 613--632 (2002; Zbl 1085.14017)] and \textit{M. Van den Bergh} [Duke Math. J. 122, No. 3, 423--455 (2004; Zbl 1074.14013)], one can consider from this data a natural perverse structure \(\mathrm{Per}(X/Y)\) on \(D^b(X)\), which is Morita-equivalent to the category of coherent modules \({\mathrm{Coh}}_A(Y)\) over a noncommutative \({\mathcal O}_Y\)-algebra \(A\). In particular \(D^b(X)\) is equivalent to \(D^b_A(Y)\).
Consider a Mukai vector \(v\) on \(X\) such that the moduli space \(Y':=M_H(v)\) has at most double point singularities. Then one can consider a vector \(w\) giving the minimal resolution \(X':=M_H(w)\), that is \(\pi': X' \to Y'\) is a birational contraction given by a nef divisor \(H'\). Then \(X \to Y\) and \(X' \to Y'\) are (twisted) Fourier-Mukai partner, and is hence natural to consider the equivalences induced by the perverse structures on both sides.
In this paper, the author considers semistable perverse sheaves on \(X\) (to be interpreted as semistable \(A\)-sheaves on \(Y\)) and carries on a detailed analysis of the correspondences given by the Fourier-Mukai duality in this situation. A particular attention is given to the case of elliptic surfaces. \(K3\) surfaces; perverse sheaves; semistable sheaves; Fourier-Mukai partners; rational double points K. Yoshioka, Perverse coherent sheaves and Fourier-Mukai transforms on surfaces I, Kyoto J. Math. 53 (2013), no. 2, 261-344. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K3\) surfaces and Enriques surfaces, Algebraic moduli problems, moduli of vector bundles, Elliptic surfaces, elliptic or Calabi-Yau fibrations Perverse coherent sheaves and Fourier-Mukai transforms on surfaces. I. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials If M is a compact, connected, simply-connected, smooth 4-manifold, and \(\gamma \in H_ 2(M; {\mathbb{Z}})\), define \(d_{\gamma}=\) minimum number of double points of immersed spheres representing \(\gamma\). Following \textit{K. Kuga} [Topology 23, 133-137 (1984; Zbl 0551.57019)], we use a theorem of \textit{S. K. Donaldson} [Bull. Am. Math. Soc., New Ser. 8, 81-83 (1983; Zbl 0519.57012)] to provide lower bounds for \(d_{\gamma}\), for \(\gamma\) certain homology classes in rational surfaces.
If \(\gamma =n\gamma _ 0+2\sum ^{r}_{i=1}\gamma _ i+\sum ^{s}_{i=r+1}\gamma _ i\in H_ 2({\mathbb{C}} {\mathbb{P}}^ 2\#m(-{\mathbb{C}} {\mathbb{P}}^ 2); {\mathbb{Z}})\), \(| n| \geq 3\), \(r<\left( \begin{matrix} | n| -1\\ 2\end{matrix} \right)\), \(s\leq m\), \(n^ 2-4r-s\geq 1\), then \(d_{\gamma}>0\). This implies \(d_{n\gamma _ 0}\geq \min (\left( \begin{matrix} | n| -1\\ 2\end{matrix} \right),[\frac{n^ 2+3}{4}]).\)
If \(\gamma =p\xi _ 1+q\xi _ 2+2\sum ^{r}_{i=1}\gamma _ i+\sum ^{s}_{i=r+1}\gamma _ i\in H_ 2(S^ 2\times S^ 2\#m(-{\mathbb{C}} {\mathbb{P}}^ 2); {\mathbb{Z}})\), \(| p|,| q| \geq 2\), \(r<(| p| -1)(| q| -1)\), \(s\leq m\), 2pq-4r-s\(\geq 1\), then \(d_{\gamma}>0\). This implies \(d_{p\xi _ 1+q\xi _ 2}\geq \min ((| p| -1)(| q| -1),[(| pq| +1)/2])\). smooth 4-manifold; double points of immersed spheres; homology classes in rational surfaces Suciu, A.: Immersed spheres in CP2 and \(S2{\times}\)S2. Math. Z. 196, 51-57 (1987) Realizing cycles by submanifolds, Immersions in differential topology, Topology of Euclidean 4-space, 4-manifolds, Surfaces and higher-dimensional varieties Immersed spheres in \({\mathbb C}\,{\mathbb P}^ 2\) and \(S^ 2\times S^ 2\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a complete list of smooth and rationally smooth normalized Schubert varieties in the twisted affine Grassmannian associated with a tamely ramified group and a special vertex of its Bruhat-Tits building. The particular case of the quasi-minuscule Schubert variety in the quasi-split but non-split form of \(\text{Spin}_8 \) (ramified triality) provides an input needed in the article by He-Pappas-Rapoport classifying Shimura varieties with good or semi-stable reduction. affine Grassmannians; Schubert varieties; intersection cohomology; local models of Shimura varieties Grassmannians, Schubert varieties, flag manifolds, Modular and Shimura varieties, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Smoothness of Schubert varieties in twisted affine Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(R\) be a discrete valuation ring and \(K\) its fraction field. Given an Abelian variety \(A_K\), we denote by \(A_K'\) its dual, by \(A,A'\) the associated Néron models and by \(\varphi,\varphi'\) their component groups. \textit{A. Grothendieck} defines [in: Sémin. Géométrie Algébrique, SGA 7 I, Exp. 9, Lect. Notes Math. 288, 313-523 (1972; Zbl 0248.14006), 1.2] a pairing \(\langle\cdot, \cdot\rangle: \varphi \times\varphi' \to\mathbb{Q}/ \mathbb{Z}\) which measures the obstruction to extending the Poincaré bundle on \(A_K\times A_K'\) as a biextension of \(A\times A'\) by \(\mathbb{G}_m\). Grothendieck's pairing is always a perfect duality for semistable Abelian varieties [\textit{A. Werner}, J. Reine Angew. Math. 486, 207-217 (1997; Zbl 0872.14037)]. For more general Abelian varieties, it is conjecturally a perfect duality if the residue field is perfect and indeed almost all has been proved in that direction, except for the case of equal positive characteristic and infinite residue field. If the residue field is not perfect, counterexamples to the perfectness of the pairing can be found [cf. \textit{A. Bertapelle} and \textit{S. Bosch}, J. Algebr. Geom. 9, 155-164 (2000; Zbl 0978.14044)].
In the present paper we prove the perfectness of Grothendieck's pairing on the \(l\)-parts of component groups when \(l\) is prime to the residue characteristic. This is sketched by \textit{A. Grothendieck} (loc. cit.; 11.3), but, as far as we know, no complete proof is to be found in the literature. Grotendieck establishes the perfectness of a similar pairing leaving it up to the reader to relate the two pairings. We show that they are equivalent up to sign. component groups of an abelian variety; discrete valuation ring; Grothendieck's pairing Bertapelle, A.: On perfectness of Grothendieck's pairing for \(l\)-parts of component groups. J. Reine Angew. Math. \textbf{538}, 223-236 (2001) Arithmetic ground fields for abelian varieties, Algebraic theory of abelian varieties, Abelian varieties of dimension \(> 1\), Local ground fields in algebraic geometry On perfectness of Grothendieck's pairing for the \(l\)-parts of component groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a criterion which characterizes a real multi-variate Laurent polynomial with full-dimensional smooth Newton polytope to have the property that all sufficiently large powers of the polynomial have fully positive coefficients. Here a Laurent polynomial is said to have fully positive coefficients if the coefficients of its monomial terms indexed by the lattice points of its Newton polytope are all positive. Our result generalizes an earlier result of Colin Tan and the author, which corresponds to the special case when the Newton polytope of the Laurent polynomial is a translate of a standard simplex. The result also generalizes a result of \textit{V. De Angelis} [Int. J. Math. Math. Sci. 2003, No. 16, 1003--1025 (2003; Zbl 1033.41016), Theorem 6.6], which corresponds to the special case of univariate polynomials. As an application, we also give a characterization of certain polynomial spectral radius functions of the defining matrix functions of Markov chains. polynomials; positive coefficients; projective toric manifolds Toric varieties, Newton polyhedra, Okounkov bodies, Real algebraic sets, Real polynomials: analytic properties, etc., Polynomials and rational functions of several complex variables, Bundle convexity Characterization of polynomials whose large powers have fully positive coefficients | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is the text of a talk given by the author in the ``automorphic semester'' held at the Centre Émile Borel at the Institut Henri Poincaré in which the author described his work (some of it with A.J. de Jong) on deformations of \(p\)-divisible groups and their Newton polygons. These results appeared in [J. Am. Math. Soc. 13, 209--241 (2000; Zbl 0954.14007)], [Ann. Math. (2) 152, 183--206, (2000; Zbl 0991.14016)] and [Prog. Math. 195, 417--440 (2001; Zbl 1086.14037)].
The main result, originally conjectured by A. Grothendieck in 1970, is that if \(G_0\) is a \(p\)-divisible group over a field \(K\) of characteristic \(p, \beta = {\mathcal N} (G_0)\) is the Newton polygon of \(G_0\) and \(\gamma\) is a Newton polygon below \(\beta\) (in the sense that every point of \(\gamma\) is on or below \(\beta\)) then there is a deformation \(G_\eta\) of \(G_0\) for which \({\mathcal N}(G_\eta) = \gamma\). There is also an analog of this result for principally polarized \(p\)-divisible groups of abelian varieties which has as a consequence a conjecture of Manin on realizing symmetric Newton polygons by abelian varieties. The author sketches some of the techniques needed to prove these results and concludes with several conjectures. An addendum (Nov. 2004) indicates that several of these are now theorems. abelian varieties; Barsotti-Tate groups; moduli spaces; Grothendieck conjecture Oort, F.: Newton polygons and p-divisible groups: a conjecture by Grothendieck. Automorphic forms I, Astérisque, No. 298, pp. 255--269. Société Mathématique de France, Paris (2005) Formal groups, \(p\)-divisible groups, Finite ground fields in algebraic geometry, Arithmetic ground fields for abelian varieties Newton polygons and \(p\)-divisible groups: a conjecture by Grothendieck. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by \textit{E. H. Kuo} [Theor. Comput. Sci. 319, No. 1--3, 29--57 (2004; Zbl 1043.05099)] and has applications to random tiling theory and the theory of cluster algebras. This work was motivated in part by the potential for applications in these areas. Additionally, we discuss an application to a problem in Donaldson-Thomas and Pandharipande-Thomas theory. The proof of our recurrence requires generalizing work of \textit{R. W. Kenyon} and \textit{D. B. Wilson} [Trans. Am. Math. Soc. 363, No. 3, 1325--1364 (2011; Zbl 1230.60009); Electron. J. Comb. 16, No. 1, Research Paper R112, 28 p. (2009; Zbl 1225.60020)]; specifically, lifting their assumption that the nodes of the graph are black and odd or white and even. double-dimer model; dimer model; condensation; recurrence Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.), Planar graphs; geometric and topological aspects of graph theory, Cluster algebras, Exact enumeration problems, generating functions, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles Combinatorics of the double-dimer model | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the moduli space \(\mathcal{R}_{3,2}\) parametrising double covers of genus 3 curves, branched along 4 distinct points. Since such a double cover is a curve of genus 7, this moduli problem may be viewed as parametrising certain generalised Prym varieties of dimension 4 which are quotients of a genus 7 Jacobian. This Prym construction provides a dominant generically finite morphism \(\mathcal{R}_{3,2} \rightarrow \mathcal{A}_4\) to Siegel space. The authors show that \(\mathcal{R}_{3,2}\) is birational to the group quotient of a product of two Grassmannian varieties. This description is similar to the descriptions obtained for various moduli spaces \(\mathcal{M}_g\) for small \(g \leq 9\) by Mukai and others. Moreover, this gives a proof of the unirationality of \(\mathcal{R}_{3,2}\) and hence a new proof for the unirationality of \(\mathcal{A}_4\). moduli spaces; curves; double covers; algebraic groups; Chow groups Structure of families (Picard-Lefschetz, monodromy, etc.), Algebraic moduli problems, moduli of vector bundles, Algebraic cycles A note on the unirationality of a moduli space of double covers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors settle two conjectures for computing Hermitian \(K\)-groups (also called Grothendieck-Witt groups) of commutative rings and of schemes. If \(R\) is a commutative ring in which \(-1\) is a sum of squares there is a homotopy equivalence
\[
BO(R)^+\sim (BGL(R)^{h\mathbb Z/2}\tag{1}
\]
which is the algebraic analogue of the homotopy equivalence \(BO\sim BU^{h\mathbb Z/2}\) between the classifying space of the orthogonal group \(O=U^{\mathbb Z/2}\subset U\) and the homotopy fixed points of \(BU\). Then there is an associated spectral sequence
\[
H^{-p}(\mathbb Z/2, K_q(R)\Rightarrow GW_{p+q}(R).
\]
When \(-1\) is not a sum of squares in \(R\) the homotopy equivalence in (1) is not valid, but its 2-adic version still holds and one gets the following homotopy fixed point theorem in terms of spectra \(\mod 2^\nu\).
Theorem 1. Let \(X\) be a \(QL\)-scheme, \(\mathcal L\) a fixed line bundle of \(X\), \(GW^{[n]}(X,\mathcal L)\) the Grothendieck-Witt spectrum of \(X\) with coefficients in the \(n\)-shifted chain complex \(\mathcal L[n]\) and \(K^{[n]}(X,\mathcal L)\) the connective \(K\)-spectrum \(K(X)\) of \(X\) equipped with the \(\mathbb Z/2\) action. Then the natural map \(GW^{[n]}(X,\mathcal L)\to K^{[n]}(X,\mathcal L)^{h\mathbb Z/2}\), between Hermitian \(K\)-theory and the homotopy fixed points of \(K\)-theory induces, for every \(\nu\geq 1\), an equivalence of spectra \(\mod 2^\nu\)
\[
GW^{[n]} (X,\mathcal L;\mathbb Z/2^\nu)\simeq K^{[n]} (X,\mathcal L;\mathbb Z/2^\nu)^{h\mathbb Z/2}.
\]
Here a scheme \(X\) is \(QL\)-scheme if it is Noetherian of finite Krull dimension, \(1/2\in\Gamma(X,\mathcal O_X)\), \(X\) has an ample family of line bundles and \(\mathrm{vcd}_2(X)=\sup\{\mathrm{vcd} 2(k(x)|x\in X\}<\infty\), where for a field \(k\) the virtual mod-2 cohomological dimension \(vcd_2(k)\) is the mod-2 etale cohomological dimension of \(k(\sqrt{-1})\).
As an application of Theorem 1 one gets the following result for complex algebraic varieties
Corollary 1. Let \(X\) be a complex algebraic variety of dimension \(d\) which has an ample family of line bundles. Let \(X(\mathbb C)\) be the associated analytic topological space of complex points. Then for \(l=2^\nu\) and \(n\in\mathbb Z\) the canonical map
\[
GW^{[n]}_i(X;\mathbb Z/l)\to KO^{2n-i}(X(\mathbb C);\mathbb Z/l)
\]
is an isomorphism for \(i\geq d-1\) and a monomorphism for \(i=d-2\). Grothendieck-Witt groups of schemes; Hermitian Quillen-Lichtenbaum conjecture; number fields; algebraic varieties Berrick, A. J.; Karoubi, M.; Schlichting, M.; Østvær, P. A., The Homotopy Fixed Point Theorem and the Quillen-Lichtenbaum conjecture in Hermitian \(K\)-theory, Adv. Math., 278, 34-55, (2015) Hermitian \(K\)-theory, relations with \(K\)-theory of rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in Hermitian \(K\)-theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We explain some remarkable connections between the two-parameter symmetric polynomials discovered in 1988 by Macdonald, and the geometry of certain algebraic varieties, notably the Hilbert scheme \(\text{Hilb}^n(\mathbb{C}^2)\) of points in the plane, and the variety \(C_n\) of pairs of commuting \(n\times n\) matrices. symmetric functions; \(n!\) conjecture; Frobenius series; diagonal harmonics; commuting variety; symmetric polynomials; algebraic varieties; Hilbert scheme M. Haiman, ''Macdonald polynomials and geometry'' in New Perspectives in Algebraic Combinatorics (Berkeley, Calif., 1996--97) , Math. Sci. Res. Inst. Publ. 38 , Cambridge Univ. Press, Cambridge, 1999, 207--254. Symmetric functions and generalizations, Research exposition (monographs, survey articles) pertaining to combinatorics, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Macdonald polynomials and geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is an improved exposition of Dwork's and Adolphson's theory of Hecke polynomials. The main technical simplification consists of the use of differential systems rather than differential operators. This avoids the necessity of ''excellent Frobenius'' and altogether the problem of ''supersingular disks''. Hecke polynomials; differential systems \(p\)-adic cohomology, crystalline cohomology, Local ground fields in algebraic geometry Les polynômes d'Hecke. Théorie p-adique. (Hecke polynomials. p-adic theory) (D'après B. Dwork et A. Adolphson) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the ``Euler characteristic integral'' of a certain cohomotopy class over its scheme of fixed points. When the base is a field and the fixed points are étale, we compute this integral in terms of Morel's identification of the ring of endomorphisms of the motivic sphere spectrum with the Grothendieck-Witt ring. In particular, we show that the Euler characteristic of an étale algebra corresponds to the class of its trace form in the Grothendieck-Witt ring. motivic homotopy theory; Grothendieck-Witt group; trace formula Hoyois, Marc, A quadratic refinement of the {G}rothendieck-{L}efschetz-{V}erdier trace formula, Algebr. Geom. Topol.. Algebraic \& Geometric Topology, 14, 3603-3658, (2014) Motivic cohomology; motivic homotopy theory, Algebraic theory of quadratic forms; Witt groups and rings, Fixed points and coincidences in algebraic topology A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0655.00010.]
This paper is an extensive survey of the known results which relate algebraic and topological K-theories to étale cohomology. There are fifteen sections to the paper, which is more than I can summarise here, in which the author explains: K-theoretic interpretations of special values of L-functions, the proof of Grothendieck's purity conjecture for \({\mathbb{Q}}_{\ell}\)-cohomology, the Riemann-Roch problem, generalisations of the Atiyah-Bott fixed point formula to algebraic groups, conjectural applications to the Tate conjecture and to the construction of a motivic cohomology theory.
The nexus of all these results is the author's main theorem which states that algebraic K-theory (with finite coefficients) of a scheme is related to étale cohomology by an Atiyah-Hirzebruch spectral sequence, after ``inverting the Bott element''. This operation coincides with the localisation of algebraic K-theory with respect to topological K-theory and it inflicts on \(mod n\quad K-theory\) a periodicity which reflects that of étale cohomology. The construction was first used by the reviewer [Mem. Ann. Math. Soc. 221 (1979; Zbl 0413.55004) and 280 (1983; Zbl 0529.55015)].
The author's point of view in applying algebraic K-theory to geometry was to push the ideas of Grothendieck and Quillen further by working not with homological algebra of rings, modules, sheaves etc. but instead to replace the algebraic objects by stable homotopy objects - ring spectra, sheaves of spectra etc. He gives a comprehensive dictionary for passing between these two universes, which should make the mentality accessible to the newcomer. étale cohomology; values of L-functions; Grothendieck's purity conjecture; Riemann-Roch problem; Atiyah-Bott fixed point formula; Tate conjecture; construction of a motivic cohomology theory; algebraic K- theory; topological K-theory; algebraic objects; stable homotopy objects Thomason, R. W.: Survey of algebraic versus étale topologicalK-theory,Algebraic K-Theory and Algebraic Number Theory, Contemporary Mathematics 83, Amer. Math. Soc., Providence (1989). Étale and other Grothendieck topologies and (co)homologies, Topological \(K\)-theory, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Riemann-Roch theorems, Applications of methods of algebraic \(K\)-theory in algebraic geometry Survey of algebraic vs. étale topological K-theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The article under review concerns computation for elliptic curves, mainly used in the field of pairing-based cryptography. The author presents a breakthrough within this area which is a new hash function on ordinary elliptic curves \(E_a:y^2=x^3+ax\) over a finite field \(\mathbb{F}_q\) whose \(j\)-invariant equals \(1728\) and satisfies the following interesting properties:
\begin{itemize}
\item it requires only one exponentiation in finite fields.
\item it can be computed in constant time.
\item it is provably indifferentiable from a random oracle.
\end{itemize}
Finally, the present contribution improves the previous fastest random oracles to \(E_a\) which perform two exponentiations in \(\mathbb{F}_q\). Calabi-Yau threefolds; double-odd curves; indifferentiable hashing to elliptic curves; \(j\)-invariant 1728; pairing-based cryptography Applications to coding theory and cryptography of arithmetic geometry, Cryptography, Curves over finite and local fields, Algebraic coding theory; cryptography (number-theoretic aspects), Elliptic curves, Rational and ruled surfaces, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Calabi-Yau manifolds (algebro-geometric aspects), \(3\)-folds, Rational and birational maps, Rationality questions in algebraic geometry, Finite ground fields in algebraic geometry The most efficient indifferentiable hashing to elliptic curves of \(j\)-invariant 1728 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\widehat{F}_2\) be the profinite completion of the free group on two generators, regarded as the algebraic fundamental group of \({\mathbb{P}}^1(\overline{\mathbb{Q}})\setminus\{0,1,\infty\}\) through the identification of generators with counter-clockwise loops around 0 and 1. There is an action of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on \(\hat{F}_2\) which leads to \textit{G. W. Anderson}'s definition of hyperadelic gamma and beta functions, \(\Gamma_\sigma\) and \(B_\sigma\), \(\sigma \in \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) [Invent. Math. 95, No. 1, 63--131 (1989; Zbl 0682.14011)]. Anderson proved a gamma factorization formula and a Gauss multiplication formula for these functions. In this paper the author generalizes the definition of these functions to \(\sigma\in \text{GT}\), the Grothendieck-Teichmüller group, which is a certain subgroup of \(\text{Aut}(\hat{F}_2)\) containing the image of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\). He uses the 5-cycle relation satisfied by elements of \(\text{GT}\) to generalize the gamma factorization formula, and formulates a generalization of the Gauss multiplication formula. He suggests that this might be essentially arithmetic, and not always satisfied for \(\sigma \in \text{GT}\), and therefore could provide a way of distinguishing a proper subgroup of \(\text{GT}\) which contains the image of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\).
For part II of this paper see \textit{Y. Ihara}, J. Reine Angew. Math. 527, 1--11 (2000; see the preceding review Zbl 1046.14009). Grothendieck-Teichmüller group; Galois theory; hyperadelic gamma and beta functions Y. Ihara, ''On beta and gamma functions associated with the Grothendieck-Teichmüller groups,'' in Aspects of Galois Theory, Cambridge, 1999, pp. 144-179. Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), \(q\)-gamma functions, \(q\)-beta functions and integrals On beta and gamma functions associated with the Grothendieck-Teichmüller group. Appendix: Profinite free differential calculus and profinite Blanchfield-Lyndon thereom | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Hyperbolic polynomials are those polynomials all of whose roots are real. The set of all monic hyperbolic polynomials of degree \(n\) is identified with a subspace \(\text{Hyp}^n\) of \({\mathbb R}^n\) (via the ordered roots with multiplicities). There is a natural stratification of \(\text{Hyp}^n\) whose closed strata \(\text{Hyp}^n_\lambda\) are indexed by number partitions \(\lambda\) of \(n\) (coming from the root multiplicities). This paper is concerned primarily with the homology of the strata of \(\text{Hyp}^n\) and lies in the area of the study of the topology of subsets of spaces of smooth maps as given by \textit{V. A. Vassiliev} [Complements of discriminants of smooth maps: topology and applications, Trans. Math. Mono., Am. Math. Soc., Providence (1994; Zbl 0826.55001)]. \textit{B. Shapiro} and \textit{V. Welker} [Result. Math. 33, No. 3-4, 338-355 (1998; Zbl 0919.57022)] have given a combinatorial construction of a finite simplicial complex \(\delta_\lambda\) whose double suspension is homeomorphic to the one point compactification of \(\text{Hyp}^n_\lambda\). The paper under review gives conditions on \(\lambda\) that imply that the one point compactification of \(\text{Hyp}^n_\lambda\) is contractible. Additional results are given concerning the homology of these compactified strata and homomorphisms of their homology groups induced by the author's notion of resonances. An ultimate goal for this line of research is to find an algorithm to compute the homology of the compactified strata. The methods are mainly combinatorial; in particular, use is made of discrete Morse theory as developed by \textit{R. Forman} [ Adv. Math. 134, No. 1, 90-145 (1998; Zbl 0896.57023)]. hyperbolic polynomials; discriminants of smooth maps; discrete Morse theory; resonances; stratification DOI: 10.1007/BF02784511 Singularities of differentiable mappings in differential topology, Stratifications in topological manifolds, Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Topology of real algebraic varieties, Discriminantal varieties and configuration spaces in algebraic topology Topology of spaces of hyperbolic polynomials and combinatorics of resonances | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove an explicit closed formula, written as a sum of Pfaffians, which describes each equivariant Schubert class for the Grassmannian of isotropic subspaces in a symplectic vector space. symplectic Grassmannian; Schubert calculus; Giambelli formula; Pfaffian; signed permutation; \(k\)-strict partition Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Equivariant Giambelli formula for the symplectic Grassmannians-Pfaffian sum formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(N\) be a positive integer and let \(\mathcal{B}_N\) be the set of all degree 2 polynomials \(f(X) =aX^2+bX+c\in{\mathbb Z}[X]\) which are not squares in \({\mathbb Z}[X]\) with the following property: There exist \(N\) consecutive integers \(r,r+1,\ldots,r+N-1\), such that \(f(r)=f(r+N-1)\) and \(f(x)\) is a square of an integer for every \(x=r,r+1,\ldots, r+N-1\). The problem of estimating the cardinality of \(\mathcal{B}_N\) for various specific values of \(N\) goes back to work of \textit{D.~Allison} [Math. Colloq., Univ. Cape Town 11, 117--133 (1977; Zbl 0371.10012), Math. Proc. Camb. Philos. Soc. 99, 381--383 (1986; Zbl 0602.10012)], as well as to work by \textit{A.~Bremner} [Acta Arith. 108, No. 2, 95--111 (2003; Zbl 1056.11033)]. In this paper, the author proves a conjecture set by A.~Bremner, namely, that \(\sharp\mathcal{B}_{10}=0\). This result, combined with the results of D.~Alison and A.~Bremner proves the following interesting:
Theorem.
\(\sharp\mathcal{B}_N=\infty\) if \(N\leq 6\) or \(N=8\) and \(\sharp\mathcal{B}_N=0\) if \(N=7\) or \(N\geq 9\).
Note that, if \(f(X)\in\mathcal{B}_N\), then, a polynomial resulting from \(f(X)\) by an integer translation of the variable \(X\) also belongs to \(\mathcal{B}_N\). Due to this simple observation, the problem of showing that \(\sharp\mathcal{B}_{10}=0\) is easily reduced to showing that there is no polynomial \(f(X)=a(X^2+X)+c\) which is not a square in \({\mathbb Z}[X]\) and has the property that \(f(i)=x_i^2\) with \(x_i \in {\mathbb Z}\) for \(i=0,\ldots,4\). On eliminating \(a\) and \(c\) from the five equations that result, one get the following system of equations, which define a genus 5 curve \(C\) in \({\mathbb P}^4\):
\[
C:\quad \begin{cases} \begin{matrix} 2x_0^2-3x_1^2+x_2^2 = 0 \\ 5x_0^2-6x_1^2+x_3^2 = 0 \\ 9x_0^2-10x_1^2+x_4^2 = 0. \end{matrix} \end{cases}
\]
Since the \(a\) appearing in \(f(X)\) is non-zero, any integral point \([x_0:x_1:x_2:x_3:x_4]\) on \(C\), furnishing an acceptable \(f(X)\), has \(x_0^2\neq x_3^2\). Setting \(t={x_1+x_3\over x_3-x_0}\) and \(s={x_3\over x_3-x_0}\) in the above equations of \(C\) and then eliminating \(s\) from them, we get the following equivalent definition of \(C\):
\[
C:\quad \begin{cases} \begin{matrix} y^2=q(t)=36t^4-72t^3+72t^2-60t=25 \\ z^2 = p(t)=(6t^2-4t-1)(6t^2+20t-25), \end{matrix} \end{cases}
\]
where \(y\) and \(z\) are also rationals. The essential part of the paper is the proof of the fact that the only rational solutions to the last system is given by \((t,y,z)=(0,\pm 5,\pm 5), ({5\over 4},\pm {25\over 8},\pm {45\over 8})\), corresponding to the projective points \([x_0:x_1:x_2:x_3:x_4]=[1:1:1:1:1], [1:3:5:7:9]\). These points, however, furnish the polynomial \(f(X)=4(X^2+X)+1\) which, being a perfect square, is not acceptable.
For the proof the author makes a detailed study of twists of a certain (explicit) unramified cover \(\chi: C'\rightarrow C\) and then he applies the so-called \textit{Elliptic Chabauty method}. This is a quite technical proof. For its general description, we quote from the paper:
``The method has two steps. Suppose we have a curve \(C\) over a number field \(K\) and an unramified map \(\chi:C' \rightarrow C\) of degree greater than one and may be defined over a finite extension \(L\) of \(K\). We consider all the distinct unramified coverings \(\chi^{(s)}:C'{}^{(s)} \rightarrow C\) formed by twists of the given one, and we get
\[
C(K)=\bigcup_{s}\chi^{(s)}(\{P\in C'{}^{(s)}(L):\chi^{(s)}(P)\in C(K)\}),
\]
the union being disjoint. Only a finite number of twists have rational points, and the finite (larger) set of twists having points locally everywhere can be explicitly described. The first step is to compute this set of twists, and the second to compute the points \(P\in C'{}^{(s)}(L)\) such that \(\chi^{(s)}(P)\in C(K)\). The second step depends on having nice quotients of the curves \(C'{}^{(s)}\), for example genus one quotients, where it is possible to do the computations.'' square values of quadratic polynomials; covering collections; elliptic Chabauty method González-Jiménez, E.; Xarles, X., On symmetric square values of quadratic polynomials, Acta Arith., 149, 2, 145-159, (2011) Counting solutions of Diophantine equations, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic ground fields for curves On symmetric square values of quadratic polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Assume \(\operatorname{char}\neq2,3\). Fix integers \(d,x,y\) such that \(d\geq 15\), \(x\geq 0\) and \(y\geq 0\). Let \(Z\subset\mathbb P^3\) be a general union of \(x\) triple points and \(y\) double points. Then either \(h^0(\mathbb P^3,{\mathcal I}_Z(d))=0\) \(\big(\)case \(10x+4y\geq\binom {d+3}{3}\big)\) or \(h^1(\mathbb P^3,{\mathcal I}_Z(d))=0\) \(\big(\)case \(10x+4y\leq \binom{d+3}{3}\big)\). polynomial interpolation; triple point; double point; zero-dimensional scheme Edoardo Ballico, On the postulation of general union of double points and triple points in P3, preprint 2007 Projective techniques in algebraic geometry On the postulation of a general union of double points and triple points in \(\mathbb{P}^3\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author gives a unified, abstract treatment of the process of setting up a category \({\mathcal D}\) of geometric objects which -- like \(C^ \infty\)-manifolds, schemes or algebraic spaces -- are constructed by gluing together spaces from a (more) basic category \({\mathcal C}\) (the category of open subspaces of Euclidean space and \(C^ \infty\)-maps in the case of \(C^ \infty\)-manifolds, and the dual of the category of rings for the other two examples). The key ingredients qualifying the category \({\mathcal C}\) for such a local structure are:
(i) A family Sub of distinguished maps called formal subsets, closed under composition and stable (the pullback of a formal subset exists and is a formal subset); the collection of formal subsets of each \(C \in {\mathcal C}\) is essentially small. Thus, schemes and algebraic spaces differ due to different choices of formal subsets in the category of affine schemes.
(ii) For each \(C \in {\mathcal C}\) a collection \(\text{Cov} (C)\) of stable effective covers of \(C\) by formal subsets (epimorphic families of formal subsets which become pullback-stable colimiting cones for their canopies, the latter term referring to the diagram obtained from the domain-objects of a cover of \(C\) by filling in the projections of the obvious pairwise fibered products over \(C)\). The system Cov is required to satisfy certain axioms, amongst which are (essentially) those for a Grothendieck topology.
A morphism of local structures or continuous functor preserves formal subsets, their pullbacks and the specified covers.
A local structure \({\mathcal C}\) (equipped with Sub and Cov) becomes a global structure if each abstract canopy or ``cut-and-paste specification'' definable in it can be realized as that of a stable effective cover of some \(C \in {\mathcal C}\). The main result states that any local structure \({\mathcal C}\) can be universally completed to a global structure \({\mathcal D}\) via a fully faithful continuous functor \({\mathcal C} \to {\mathcal D}\) by iterating (twice) a certain ``plus-construction''. The classical examples (including rigid analytic spaces and Douady's espaces analytique banachique) fit in this mould, and can thus be usefully recognized as universal constructions. \(C^ \infty\)-manifolds; cut-and-paste specification; plus-construction; Banach analytic spaces; schemes; algebraic spaces; local structure; formal subsets; pullback-stable; canopies; Grothendieck topology; continuous functor; global structure; rigid analytic spaces; espaces analytique banachique; universal constructions Feit P., Axiomization of Passage from 'Local' Structure to 'Global' Object 91 pp 485-- (1993) Grothendieck topologies and Grothendieck topoi, Abstract manifolds and fiber bundles (category-theoretic aspects), Generalizations (algebraic spaces, stacks), Research exposition (monographs, survey articles) pertaining to category theory, Topoi, Banach analytic manifolds and spaces Axiomization of passage from ``local'' structure to ``global'' object | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a case of a positivity conjecture of
\textit{L. C. Mihalcea} and \textit{R. Singh} [``Mather classes and conormal spaces of Schubert varieties in cominuscule spaces'', Preprint, \url{arXiv:2006.04842}],
concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassmannian \(LG(n,2n)\). Combined with work of
\textit{P. Aluffi} et al. [``Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells'', Preprint, \url{arXiv:1709.08697}],
this further implies the positivity of the Mather classes for Schubert varieties in \(LG(n,2n)\), which Mihalcea-Singh had verified for the other cominuscule spaces of classical Lie type. Building on the work of Boe and Fu, we give a positive recursion for the local Euler obstructions, and use it to show that they provide a positive count of admissible labelings of certain trees, analogous to the ones describing Kazhdan-Lusztig polynomials. Unlike in the case of the Grassmannians in types A and D, for \(LG(n,2n)\) the Euler obstructions \(e_{y,w}\) may vanish for certain pairs \((y,w)\) with \(y\le w\) in the Bruhat order. Our combinatorial description allows us to classify all the pairs \((y,w)\) for which \(e_{y,w}=0\). Restricting to the big opposite cell in \(LG(n,2n)\), which is naturally identified with the space of \(n\times n\) symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification. local Euler obstructions; Schubert stratification; Lagrangian Grassmannian; tree labelings Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Trees, Local complex singularities, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) Euler obstructions for the Lagrangian Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The universal Abel map \(\mathcal M_{g,n} \rightarrow \mathcal J_{g,n}\) extends in general only to a rational map \(\overline{ \mathcal M}_{g,n} \dashrightarrow \overline {\mathcal J}_{g,n}\) from the Deligne-Mumford compactification of the moduli space of smooth curves to a compactification of the universal Jacobian given by some choice of universal stability condition. Roughly speaking, the issue is that the line bundles one obtains as images of the Abel map need not be stable, and a stable representative depends on the choice of a one-parameter smoothing of the curve. There are two natural approaches to this issue: first, one can modify \(\overline{ \mathcal M}_{g,n}\) to resolve the indeterminancy as for example in [\textit{D. Holmes}, J. Inst. Math. Jussieu 20, No. 1, 331--359 (2021; Zbl 1462.14031)] or in [\textit{S. Marcus} and \textit{J. Wise}, Proc. Lond. Math. Soc. (3) 121, No. 5, 1207--1250 (2020; Zbl 1455.14021)]; or second, one can tailor a stability condition to obtain a compactified Jacobian that avoids the issue for a given Abel map as in [\textit{J. L. Kass} and \textit{N. Pagani}, Trans. Am. Math. Soc. 372, No. 7, 4851--4887 (2019; Zbl 1423.14187)].
In this paper, the authors follow the first approach and describe a blow-up of \(\overline{ \mathcal M}_{g,n}\) that resolves the indeterminancy of the universal Abel map. This resolution is formulated in terms of tropical geometry. Namely, the tropical universal Abel map is not a morphism of generalized cone complexes, for the analogous reason as in the algebro-geometric setting: given a divisor on a tropical curve, there is a unique stable representative linearly equivalent to it, but this representative depends on the edge lengths of the underlying graph. Refining the cone structure of the moduli space of tropical curves turns the tropical universal Abel map into a morphism of cone complexes, which describes the desired blow-up of \(\overline{ \mathcal M}_{g,n}\) by a standard construction of toric geometry. Much of the tropical analysis is done in the authors' previous work [\textit{A. Abreu} and \textit{M. Pacini}, Proc. Lond. Math. Soc. (3) 120, No. 3, 328--369 (2020; Zbl 1453.14082)], to which the current paper serves as an algebro-geometric counterpart. As an application, the authors give descriptions of the algebro-geometric and tropical double ramification cycles. geometric Abel map; tropical Abel map; double ramification cycle Families, moduli of curves (algebraic), Jacobians, Prym varieties, Applications of tropical geometry The resolution of the universal Abel map via tropical geometry and applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(C\) be a smooth projective complex curve. The gonality \(\text{gon}(C)\) of \(C\) is defined as the smallest integer \(d\) such that there exists a free linear series of degree \(d\) on \(C\). Another invariant of \(C\), denoted by \(d_0(C)\), is the smallest integer \(d\) such that for every \(m\geq d\) there exists on \(C\) a free linear series of degree \(d\). In the paper under consideration, the author investigates the behaviour of \(d_0(C)\), with special attention to the case in which there exists a degree 2 map \(f:C\to C'\) onto a curve \(C'\). linear system; double cover; gonality Keem, Changho, Double coverings of smooth algebraic curves, (Algebraic Geometry in East Asia, Kyoto, 2001, (2002), World Sci. Publ. River Edge, NJ), 75-111 Special divisors on curves (gonality, Brill-Noether theory), Coverings of curves, fundamental group Double coverings of smooth algebraic curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a complex vector space equipped with a nondegenerate symmetric bilinear form. Let \(X\) denote the flag variety for the even orthogonal group, which parameterizes flags of isotropic subspaces in \(V\). In the paper under review, the author develops a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of \(X\). These polynomials are applied to understand the structure of the Gillet-Soulé arithmetic Chow ring of \(X\). Actually, the author studies an extra relation which comes from the vanishing of the top Chern class of the maximal isotropic subbundle of the trivial vector bundle over \(X\) and he computes the natural arithmetic Chern numbers on \(X\). Finally, he shows that these arithmetic Chern numbers are all rational. Schubert polynomial; orthogonal flag variety; arithmetic Chow ring; arithmetic Chern number Harry Tamvakis, Schubert polynomials and Arakelov theory of orthogonal flag varieties, Math. Z. 268 (2011), no. 1-2, 355 -- 370. Grassmannians, Schubert varieties, flag manifolds, Arithmetic varieties and schemes; Arakelov theory; heights, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Schubert polynomials and Arakelov theory of orthogonal flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials What is the maximum number of common zeros that a system of \(r\) linearly independent homogeneous polynomials of degree \(d\) in \(\mathbb{F}_q[x_0,x_1,\ldots, x_m]\) can have in \(\mathbb{P}^m(\mathbb{F}_q)\)?
The \(r=1\) case was settled by \textit{J.-P. Serre} [in: Journées arithmétiques. Exposés présentés aux seizièmes congrès en Luminy, France, 17-21 Juillet, 1989. Paris: Société Mathématique de France. 351--353 (1991; Zbl 0758.14008)], and independently by \textit{A. B. Sørensen} [IEEE Trans. Inf. Theory 37, No. 6, 1567--1576 (1991; Zbl 0741.94016)], where the corresponding varieties are hyperplanes that intersect in a linear subspace of codimension \(2\). The \(r=2\) case is due to \textit{M. Boguslavsky} [Finite Fields Appl. 3, No. 4, 287--299 (1997; Zbl 0924.11097)]. A problem equivalent to the \(m=2\) case was solved by \textit{C. Zanella} [Des. Codes Cryptography 13, No. 2, 199--212 (1998; Zbl 0895.94011)]. Boguslavsky and Tsfasman gave a general conjecture, but the authors of this paper showed that it does not hold in some cases where \(r> m+1\) and \(d>1\) [Contemp. Math. 686, 157--169 (2017; Zbl 1367.14008)].
The main result of this paper is the following.
Theorem. Assume that \(1 \leq d \leq q-1\) and \(1 \leq r \leq m+1\). The the maximum number of common zeros in \(\mathbb{P}^m(\mathbb{F}_q)\) that a system of \(r\) linearly independent homogeneous polynomials, each of degree \(d\) in \(S = \mathbb{F}_q[x_0,x_1,\ldots, x_m]\), can have is given by
\[
\begin{cases} (q^{m-r+1}-1)/(q-1) & \text{if } d=1 \text{ and } 1\leq r \leq m+1,\\ (d-1) q^{m-r} + (q^{m-1}-1)/(q-1) + \lfloor q^{m-r} \rfloor& \text{if } d>1 \text{ and } 1\leq r \leq m+1. \end{cases}
\]
Moreover, this bound is always attained. Furthermore, if \(d>1\) and if the maximum is attained by a family \(\{F_1,\ldots, F_r\}\) of \(r\) linearly independent polynomials in \(S\), each of degree \(d\), then \(F_1,\ldots, F_r\) must have a common linear factor.
The proof uses an affine analogue due to \textit{P. Heijnen} and \textit{R. Pellikaan} [IEEE Trans. Inf. Theory 44, No. 1, 181--196 (1998; Zbl 1053.94581)]. This affine result leads to an easy proof of the case where the polynomials share a common linear factor.
The proof of the theorem is divided into cases based on the degrees of the greatest common divisors of pairs of the polynomials. In the most difficult case this degree is \(d-1\) for every pair, and here the authors use results on \textit{coprime close families} of polynomials developed in [\textit{S. R. Ghorpade} and \textit{G. Lachaud}, Finite Fields Appl. 7, No. 4, 468--506 (2001; Zbl 1007.94024)]. The authors update the Tsfasman-Boguslavsky conjecture, giving a proposed answer for all \(d\leq q\) and \(r \leq \binom{m+d-1}{m}\). systems of polynomial equations; polynomials over finite fields; Tsfasman-Boguslavsky conjecture Datta, M.; Ghorpade, S. R., Number of solutions of systems of homogeneous polynomial equations over finite fields, Proc. Amer. Math. Soc., 145, 2, 525-541, (2017) Finite ground fields in algebraic geometry, Polynomials over finite fields, Varieties over finite and local fields, Rational points, Combinatorial structures in finite projective spaces, Combinatorial aspects of finite geometries Number of solutions of systems of homogeneous polynomial equations over finite fields | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An effective algorithm for a smooth (weak) stratification of a real semi- Pfaffian set is suggested, provided an oracle deciding consistency of a system of Pfaffian equations and inequalities is given. An explicit estimate of the complexity of the algorithm and of the resulting stratification is given, in terms of the parameters of the Pfaffian functions defining the original semi-Pfaffian set. The algorithm is applied to sets defined by sparse polynomials and exponential polynomials. semi-Pfaffian set; sparse polynomials; exponential polynomials Gabrièlov, A.; Vorobjov, N., Complexity of stratifications of semi-Pfaffian sets, Discrete Comput. Geom., 14, 71-91, (1995) Analysis of algorithms and problem complexity, Real-analytic and semi-analytic sets Complexity of stratifications of semi-Pfaffian sets | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials All possible bracketings of \(n\) symbols in all possible orders are exhibited as vertices of a combinatorial CW-complex \(KP_ n\). It is clearly relevant to the coherence of symmetric monoidal categories, yet also fits nicely into Drinfel'd's study of the Knizhnik-Zamolodchikov equations and into the analysis of the Grothendieck-Knudsen moduli space of stable \(n\)-pointed curves of genus 0. associahedron; braided tensor category; coherence; symmetric monoidal categories; Knizhnik-Zamolodchikov equations; Grothendieck-Knudsen moduli space M.M. Kapranov, \textit{The permutoassociahedron, Mac Lane}'\textit{s coherence theorem and asymptotic zones for the KZ equation}, \textit{J. Pure Appl. Alg.}\textbf{85} (1993) 119. Monoidal categories (= multiplicative categories) [See also 19D23], Arithmetic ground fields (finite, local, global) and families or fibrations, Partial differential equations of mathematical physics and other areas of application, Symmetric monoidal categories The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be an algebraically closed field of characteristic \(p>0\) and \(C\) a connected nonsingular projective curve over \(k\) with genus \(g\geq 2\). Let \((C,G)\) be a ``big action,'' i.e. a pair \((C,G)\) where \(G\) is a \(p\)-subgroup of the \(k\)-automorphism group of \(C\) such that \(\frac{|G|}{g}> \frac{2p}{p-1}\). The aim of this paper is to describe the big actions whose derived group \(G{^{\prime}}\) is \(p\)-elementary abelian. In particular, we obtain a structure theorem for the functions parametrizing the Artin-Schreier cover \(C\rightarrow C/G{^{\prime}}\). Using Artin-Schreier duality, we shift to a group-theoretic point of view to characterize relevant cases. Then, we display universal families and discuss the corresponding deformation space for \(p=5\). automorphism of curves; families of curves; Artin-Schreier covers; special groups; additive polynomials Rocher, M., Large \(p\)-group actions with a \(p\)-elementary abelian derived group, J. Algebra, 321, 2, 704-740, (2009) Automorphisms of curves Large \(p\)-group actions with a \(p\)-elementary abelian derived group | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We estimate the expected value of the gradient degree of certain Gaussian random polynomials in two variables and discuss its relations with some other numerical invariants of random polynomials. expected value of the gradient degree; Gaussian random polynomials in two variables T. Aliashvili, ''Topological invariants of random polynomials,'' In: Banach Center Publ., 62, 19--28 (2004). Geometric probability and stochastic geometry, Topology of real algebraic varieties On invariants of random planar endomorphisms | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is largely a survey paper. Its focus is on various Grothendieck topologies on affine schemes and their connections to recent works in commutative algebra surrounding the direct summand theorem and the existence of big Cohen-Macaulay algebras. In the first part, the authors review basics of Grothendieck topology, with an emphasize on the canonical topology on affince schemes, including a proof that the canonical topology on affine schemes is equivalent to the (effective) descent topology (which was originally due to Olivier). In the second part, the authors give a geometric interpretation of the direct summand theorem (resp. the existence of big Cohen-Macaulay algebras). Namely, the direct summand theorem (resp. existence of big Cohen-Macaulay algebras) is saying that any finite cover of a regular affine scheme is a covering for the canonical topology (resp. the fpqc topology). In the third part, the authors survey what was known about splinters, i.e., rings that splits off from all their module-finite extensions, including the derived variants and connections to F-singularities. Some interesting questions about an fpqc analog of splinters are raised. Examples are given throughout the article, including a negative answer to a question of Ferrand (Example 10.6). Grothendieck topology; canonical cover; fpqc cover; direct summand theorem; big Cohen-Macaulay algebras; splinters Relevant commutative algebra, Schemes and morphisms, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Grothendieck topologies and Grothendieck topoi, Extension theory of commutative rings, Morphisms of commutative rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry On the canonical, fpqc, and finite topologies on affine schemes. The state of the art | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth projective complex surface. The author introduces analogues of Donaldson polynomials, christened ``almost canonical spin polynomials'': They are defined by intersecting certain ``divisors'' on the moduli space of (stable) rank two vector bundles \(V\) on \(X\) with \(c_1(V)= c_1(K_X)\), \(c_2(V)= c_2(X)+k\), and \(h^0(V)>0\). The main result is the so called shape theorem, which asserts that these spin polynomials are expressible in terms of the intersection form, the canonical class, and the Poincaré duals of certain curves on \(X\). Donaldson polynomials; spin polynomials Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Special surfaces Canonical and almost canonical spin polynomials of an algebraic surface | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this fundamental paper the author, after a long period of working, proved the Grothendieck conjecture, a very deep and difficult result concerning the converse of the Grothendieck specialization theorem for \(p\)-divisible groups -- the Newton polygon goes up under specialization in a family of \(p\)-divisible groups [\textit{A. Grothendieck}, Groupes de Barsotti-Tate et cristaux de Dieudonné (Les Presses de l'Universite de Montreal) (1974; Zbl 0331.14021)]. The analogous statement for the reduction mod \(p\) of Siegel moduli spaces is also proved. The proof relies on two rather different aspects of deformation theory of \(p\)-divisible groups. The first one is on deformations of simple \(p\)-divisible groups keeping the Newton polygon constant. The author uses the methods and results derived from ``Purity'' as obtained in [\textit{A. J. de Jong} and \textit{F. Oort}, J. Am. Math. Soc. 13, 209--241 (2000; Zbl 0954.14007)] . The other one is an explicit deformation theory with invariant \(a\)-number one [\textit{F. Oort}, Ann. Math. (2) 152, 183--206 (2000; Zbl 0991.14016)]. This is an effective method of reading the Newton polygon of a subvariety in a local deformation space. In this paper a combination of these two methods gives the desired result. \(p\)-divisible groups; Newton polygons; Grothendieck conjecture F. Oort: Newton polygon strata in the moduli space of abelian varieties, Moduli of abelian varieties (Texel Island, 1999), 417--440, Progr. Math. 195, Birkhäuser, Basel, 2001. Algebraic moduli of abelian varieties, classification, Formal groups, \(p\)-divisible groups Newton polygon strata in the moduli space of abelian varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We define Schubert Eisenstein series as sums like usual Eisenstein series but with the summation restricted to elements of a particular Schubert cell, indexed by an element of the Weyl group. They are generally not fully automorphic. We will develop some results and methods for \(\mathrm{GL}_3\) that may be suggestive about the general case. The six Schubert Eisenstein series are shown to have meromorphic continuation and some functional equations. The Schubert Eisenstein series \(E_{s_1s_2}\) and \(E_{s_2s_1}\) corresponding to the Weyl group elements of order three are particularly interesting: at the point where the full Eisenstein series is maximally polar, they unexpectedly become (with minor correction terms added) fully automorphic and related to each other. Schubert Eisenstein series; Schubert cell; meromorphic continuation; Weyl group Other groups and their modular and automorphic forms (several variables), Grassmannians, Schubert varieties, flag manifolds Schubert Eisenstein series | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We examine a suggested relation between stochastic quantization and the holographic Wilsonian renormalization group in the massive fermion case on Euclidean AdS space. The original suggestion about the general relation between the two theories is posted in [\textit{J.-H. Oh} and \textit{D. P. Jatkar}, ``Stochastic quantization and holographic Wilsonian renormalization group'', Preprint, \url{arXiv:1209.2242}]. In the previous researches, it is already verified that scalar fields, \(\mathrm{U}(1)\) gauge fields, and massless fermions are consistent with the relation. In this paper, we examine the relation in the massive fermion case. Contrary to the other case, in the massive fermion case, the action needs particular boundary terms to satisfy boundary conditions. We finally confirm that the proposed suggestion is also valid in the massive fermion case. stochastic quantization; holographic Wilsonian renormalization group; fermion; double trace deformation Quantization in field theory; cohomological methods, Stochastic quantization, Renormalization group methods applied to problems in quantum field theory, Formal methods and deformations in algebraic geometry, Commutation relations and statistics as related to quantum mechanics (general) Stochastic quantization and holographic Wilsonian renormalization group of free massive fermion | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Dans l'anneau gradué \(A=k[x_0, \dots, x_n]\), \(A= \bigoplus^\infty_{\alpha=0} A_\alpha\), soit \(I= \bigoplus^\infty_{\alpha=0} I_\alpha\) \((I_\alpha =I\cap A_\alpha)\) un idéal homogène; posons \(v_I(l)= \dim_k(I_l/I_{l-1} A\cap A_\alpha)\). Indiquons avec \({\mathcal I} (l_1, \dots, l_q;n)\) (où \(l_1= \cdots =l_{m_1} <l_{m_1+1} =\cdots =l_{m_2}< \cdots< l_{m_p+1} = \cdots =l_q\) est une suite donnée de nombres entiers positifs) la famille des idéaux homogènes \(I\) de \(A\) pour lesquels
\[
v_I(l)= \begin{cases} 0 \quad & \text{si } 0\leq l\leq l_1, \\ m_1 \quad & \text{si } l=l_1= \cdots= l_{m_1}, \\ 0 \quad & \text{si } l_{m_1} <l<l_{m_1+1}, \\ m_2-m_1 \quad & \text{si } l=l_{m_1+1} =\cdots= l_{m_2}, \\ 0 \quad & \text{si } l=l_{m_2} <l< l_{m_2+1}, \\ m_3-m_2 \quad & \text{si } l=l_{m_2+1} =\cdots=l_{m_3}, \\ \cdots\\ 0 \quad & \text{si } l_{m_p} <l<l_{m_p+1}, \\ q-m_p \quad & \text{si } l=l_{m_p+1} =\cdots =l_q, \\ 0 \quad & \text{si } l>l_q. \end{cases}
\]
Proposition. \({\mathcal I} (l_1, \dots, l_q;n)\) a une structure de variété algébrique.
Proposition. La projection canonique \(p\): \(\Omega\to {\mathcal I} (l_1, \dots, l_q;n)\) est une application ouverte.
Proposition. \({\mathcal H} (l_1, \dots, l_q;n)\) a une structure de variété algébrique, en correspondance biunivoque avec un ouvert de Zariski de \({\mathcal I} (l_1, \dots, l_q;n)\).
[For part II of this paper see \textit{C. Perelli Cippo}, same volume, Banach Cent. Publ. 37, 71-74 (1996; see the following review)]. ideals of polynomials; exterior differential forms Polynomial rings and ideals; rings of integer-valued polynomials, Exterior differential systems (Cartan theory), Ideals and multiplicative ideal theory in commutative rings, Relevant commutative algebra Remarks on ideals of polynomials and of exterior differential forms. I | 0 |
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