Quadric Hypersurface Research Articles

On the Injectivity of Euler Integral Transforms with Hyperplanes and Quadric Hypersurfaces

On the Injectivity of Euler Integral Transforms with Hyperplanes and Quadric Hypersurfaces

Clifford Deformations of Koszul Frobenius Algebras and Noncommutative Quadrics

A Clifford deformation of a Koszul Frobenius algebra [Formula: see text] is a finite dimensional [Formula: see text]-graded algebra [Formula: see text], which corresponds to a noncommutative quadric hypersurface [Formula: see text] for some central regular element [Formula: see text]. It turns out that the bounded derived category [Formula: see text] is equivalent to the stable category of the maximal Cohen-Macaulay modules over [Formula: see text] provided that [Formula: see text] is noetherian. As a consequence, [Formula: see text] is a noncommutative isolated singularity if and only if the corresponding Clifford deformation [Formula: see text] is a semisimple [Formula: see text]-graded algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces. As an application, we recover Knörrer's periodicity theorem without using matrix factorizations.

Hofer geometry via toric degeneration

The main theme of this paper is to use toric degeneration to study Hofer geometry by producing distinct homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. We focus on the (complex n-dimensional) quadric hypersurface and the del Pezzo surfaces, and study two classes of distinguished Lagrangian submanifolds that appear naturally in a toric degeneration, namely the Lagrangian torus which is the monotone fiber of a Lagrangian torus fibration, and the Lagrangian spheres that appear as vanishing cycles. For the quadrics, we prove that the group of Hamiltonian diffeomorphisms admits two distinct homogeneous quasimorphisms and derive some superheaviness results. Along the way, we show that the toric degeneration is compatible with the Biran decomposition. This implies that for n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=2$$\\end{document}, the Lagrangian fiber torus (Gelfand–Zeitlin torus) is Hamiltonian isotopic to the Chekanov torus, which answers a question of Y. Kim. We prove analogous results for the del Pezzo surfaces. We also discuss applications to C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^0$$\\end{document} symplectic topology.

Dynamics of Chebyshev endomorphisms on some affine algebraic varieties

The Chebyshev polynomials Td in one variable are typical chaotic maps on C. Chebyshev endomorphisms PAnd: Cn→Cn are also chaotic. We consider the action of the dihedral group Dn+1 on Cn. The endomorphism PAnd maps any Dn+1-orbit of z∈Cn to a Dn+1-orbit of PAnd(z). The endomorphism PAnd induces a mapping on Cn∕Dn+1. Using invariant theory, we embed Cn∕Dn+1 as an affine subvariety X in Cm. Then we have morphisms gd on X. We study the cases n=2 and 3. In these cases, the morphisms gd are defined over Z. We find a class of affine subvarieties V of X which are invariant under gd. These varieties are concerned with branch loci or critical loci. The class contains C2, a cuspidal cubic, a parabola, a quadric hypersurface in C4, an affine algebraic surface in C4 which is birationally equivalent to an affine quadric cone in C3, and others. For each affine variety V in the class, there exists a polynomial parameterization PV satisfying gd|V(PV(y1,…,yk))=PV(Td(y1),…,Td(yk)), where Td(z) is a Chebyshev polynomial in one variable. Then we determine the set of bounded orbits of gd|V in each invariant set V and give relations between them.

The Hilbert-Kunz density functions of quadric hypersurfaces

We show that the Hilbert-Kunz density function of a quadric hypersurface of Krull dimension n+1 is a piecewise polynomial on a subset of [0,n], whose complement in [0,n] has measure zero. Our explicit description of the Hilbert-Kunz density function confirms a conjecture of Watanabe-Yoshida on the lower bound of the Hilbert-Kunz multiplicity of the quadric of dimension n+1, provided the characteristic is at least n−1. We also show that the Hilbert-Kunz multiplicity of a quadric of fixed dimension is an eventually strictly decreasing function of the characteristic confirming a conjecture of Yoshida.The main input comes from the classification of Arithmetically Cohen-Macaulay bundles on the projective variety defined by the quadric via matrix factorizations.

Geometry of elliptic normal curves of degree 6

In our work we focus on the geometry of elliptic normal curves of degree 6 embedded in . We determine the space of quadric hypersurfaces through an elliptic normal curve of degree 6 and find the explicit equations of generators of . We study the images and of a sextic under the projection from a general point and a general line . In particular, we show that is k-normal for all and is generated by three homogeneous polynomials of degree 2 and two homogeneous polynomials of degree 3. We then show that is k-normal for all and is generated by two homogeneous polynomials of degree 3 and three homogeneous polynomials of degree 4.

Iterated and mixed discriminants

Classical work by Salmon and Bromwich classified singular intersections of two quadric surfaces. The basic idea of these results was already pursued by Cayley in connection with tangent intersections of conics in the plane and used by Schäfli for the study of hyperdeterminants. More recently, the problem has been revisited with similar tools in the context of geometric modeling and a generalization to the case of two higher dimensional quadric hypersurfaces was given by Ottaviani. We propose and study a generalization of this question for systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant . We define a related polynomial called the multivariate iterated discriminant , generalizing the classical Schäfli method for hyperdeterminants. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre–Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant.

Kernel-Free Nonlinear Support Vector Machines for Multiview Binary Classification Problems

Multiview learning (MVL) frequently uses support vector machine- (SVM-) based models, but it can be difficult to select appropriate kernel functions and corresponding parameters. Then, by introducing kernel-free tricks, two multiview classifiers are proposed, called C -multiview kernel-free nonlinear support vector machine ( C -MKNSVM) and its ν -version, namely, ν -MKNSVM. They try to find a quadratic hypersurface under each view to classify the sample points and employ a consistency constraint to fuse the sample points from two views. Both the primal and dual problems of C -MKNSVM and ν -MKNSVM do not involve kernel functions; thus, they are allowed to be solved directly. In addition, the relationship of solutions between the primal and dual problems is discussed in each classifier. For the C -version and ν -version of MKNSVM, the meanings of their parameters and the relationship between them are analyzed in detail. The experimental results of artificial and benchmark datasets show that our methods are superior to some traditional MVL classifiers like SVM-2K, PSVM-2V, and MvTSVM, especially ν -MKNSVM.

Disk potential functions for quadrics

We compute the disk potential of Gelfand–Zeitlin monotone torus fiber in a quadric hypersurface by exploiting toric degenerations, Lie theoretical mirror symmetry, and the structural result of the monotone Fukaya category.

Noncommutative conics in Calabi-Yau quantum projective planes

In noncommutative algebraic geometry, noncommutative quadric hypersurfaces are major objects of study. In this paper, we focus on studying noncommutative conics ProjncA embedded into Calabi-Yau quantum projective planes. In particular, we give complete classifications of homogeneous coordinate algebras A of noncommutative conics up to isomorphism of graded algebras, and of noncommutive conics ProjncA up to isomorphism of noncommutative schemes.

An Identity Theorem for the Fourier–Laplace Transform of Polytopes on Nonzero Complex Multiples of Rationally Parameterizable Hypersurfaces

A set mathcal {S} of points in mathbb {R}^n is called a rationally parameterizable hypersurface if there is vector function varvec{sigma }:mathbb {R}^{n-1}rightarrow mathbb {R}^n having as components rational functions defined on some common domain D such that mathcal {S}={varvec{sigma }(textbf{t}):textbf{t}in D}. A generalized n-dimensional polytope in mathbb {R}^n is a union of a finite number of convex n-dimensional polytopes in mathbb {R}^n. The Fourier–Laplace transform of such a generalized polytope mathcal {P} in mathbb {R}^n is defined by F_{mathcal {P}}(textbf{z})=int _{mathcal {P}}e^{textbf{z}cdot textbf{x}},textbf{dx}. Let gamma be a fixed nonzero complex number. We prove that F_{mathcal {P}_1}(gamma varvec{sigma }(textbf{t}))=F_{mathcal {P}_2}(gamma varvec{sigma }(textbf{t})) for all textbf{t} in O implies mathcal {P}_1=mathcal {P}_2 if O is an open subset of D satisfying some well-defined conditions and we present similar results for the null set of the Fourier–Laplace transform of mathcal {P}. Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres.

Foliations by curves on threefolds

AbstractWe study the conormal sheaves and singular schemes of one‐dimensional foliations on smooth projective varieties X of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is μ‐stable whenever the tangent bundle is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on and on a smooth quadric hypersurface . Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q3.

Derived categories of skew quadric hypersurfaces

The existence of a full strong exceptional sequence in the derived category of a smooth quadric hypersurface was proved by Kapranov. In this paper, we present a skew generalization of this result. Namely, we show that if S is a standard graded (±1)-skew polynomial algebra in n variables with n ≥ 3 and f = x 21 + ⋯ + x 2n ∈ S, then the derived category Db(qgr S/(f)) of the noncommutative scheme qgr S/(f) has a full strong exceptional sequence. The length of this sequence is given by n−2+2r where r is the nullity of a certain matrix over \({\mathbb{F}_2}\). As an application, by studying the endomorphism algebra of this sequence, we obtain the classification of Db(qgr S/(f)) for n = 3, 4.

The Finiteness of Smooth Curves of Degree $\le 11$ and Genus $\le 3$ on a General Complete Intersection of a Quadric and a Quartic in $\mathbb{P}^5$

Let $W\subset \mathbb{P}^5$ be a general complete intersection of a quadric hypersurface and a quartic hypersurface. In this paper, we prove that $W$ contains only finitely many smooth curves$C\subset \mathbb{P}^5$ such that $d:= \deg ({C}) \le 11$, $g:= p_a({C}) \le 3$ and $h^1(\mathcal{O} _C(1)) =0$.

Gamma conjecture II for quadrics

The Gamma conjecture II for the quantum cohomology of a Fano manifold F, proposed by Galkin, Golyshev and Iritani, describes the asymptotic behavior of the flat sections of the Dubrovin connection near the irregular singularities, in terms of a full exceptional collection, if there exists, of \({\mathcal {D}}^b(F)\) and the \({\widehat{\Gamma }}\)-integral structure. In this paper, for the smooth quadric hypersurfaces we prove the convergence of the full quantum cohomology and Gamma II. For the proof, we first give a criterion on Gamma II for Fano manifolds with semisimple quantum cohomology, by Dubrovin’s theorem of analytic continuations of semisimple Frobenius manifolds. Then we work out a closed formula of the Chern characters of spinor bundles on quadrics. By the deformation-invariance of Gromov–Witten invariants we show that the full quantum cohomology can be reconstructed by its ambient part, and use this to obtain estimations. Finally we complete the proof of Gamma II for quadrics by explicit asymptotic expansions of flat sections corresponding to Kapranov’s exceptional collections and an application of our criterion.

Prediction of microbial growth via the hyperconic neural network approach

The design and control of biological reactors depends upon the correct calibration and selection of kinetic parameters from the exponential phase in the microbial growth. The kinetic microbial parameters that are reported in the literature of biochemical engineering are obtained based on specific growth conditions, which limits their potential application in a wide variety of dynamic models. In this paper, we propose an adaptive machine-learning approach to predict microbial growth, and provide information on the effect that variations of pH and concentration of the culture media have on the growth rate. We study biological reactions and obtain a set of experimental exploratory data using the Pseudomonas aeruginosa (i.e., P. aeruginosa). The versatile and robust metabolism of P. aeruginosa is responsible for its ability to grow in different environment conditions, even at low nutrient and oxygen levels, in a sample range of temperatures (4∘-42∘C) and polluted sites. Our first contribution is to propose a technique to gather experimental data via measurements of optical density, microbial growth, from the Multiskan™ FC Microplate Photometer, Thermo Scientific. Our second contribution consists of building a surrogate model by successfully adapting the Hyperconic Multilayer Perceptron (HC-MLP), a novel computational and mathematical approach, to predict microbial growth across all the set of conditions of the experimental design. HC-MLP is a state-of-the-art method that is used to define complex non-linear decision boundaries, in the parameters’ space, using a mix of ideas from conformal geometry and neural networks, and focusing on quadratic hyper-surfaces through multiple hidden layers. Consequently, we generate precise surrogates, which can even predict microbial growth in values not evaluated during the experimental stage. Finally, our statistical testing and comparisons validate that the proposed experimental, mathematical and computational framework is robust and capable of predicting the dynamic growth of bacteria P. aeruginosa using two main operation conditions: pH and concentration culture media at pH 7.0. In the future, we plan to apply our proposed methodology to other bacterial strains and advance HC-MLP for forecasting dynamics of other multiple microbial measurements under a wide variety of conditions.

Integral points on affine quadric surfaces

It is well-known that the Hasse principle holds for quadric hypersurfaces. The Hasse principle fails for integral points on smooth quadric hypersurfaces of dimension 2 but the failure can be completely explained by the Brauer-Manin obstruction. We investigate how often the family of quadric hypersurfaces $ax^2 + by^2 +cz^2 = n$ has a Brauer-Manin obstruction. We improve previous bounds of Mitankin.

Homogeneous quasimorphisms, $C^0$-topology and Lagrangian intersection

We construct an example of a non-trivial homogeneous quasimorphism on the group of Hamiltonian diffeomorphisms of the two and four dimensional quadric hypersurfaces which is continuous with respect to both the C^0 -metric and the Hofer metric. This answers a variant of a question of Entov–Polterovich–Py which is one of the open problems listed in the monograph of McDuff–Salamon. Throughout the proof, we make extensive use of the idea of working with different coefficient fields in quantum cohomology rings. As a by-product of the arguments in the paper, we answer a question of Polterovich–Wu regarding homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms of the complex projective plane and prove some intersection results about Lagrangians in the four dimensional quadric hypersurface.

Noncommutative Knörrer’s periodicity theorem and noncommutative quadric hypersurfaces

Noncommutative hypersurfaces, in particular, noncommutative quadric hypersurfaces are major objects of study in noncommutative algebraic geometry. In the commutative case, Kn\"orrer's periodicity theorem is a powerful tool to study Cohen-Macaulay representation theory since it reduces the number of variables in computing the stable category $\underline{\operatorname{CM}}(A)$ of maximal Cohen-Macaulay modules over a hypersurface $A$. In this paper, we prove a noncommutative graded version of Kn\"orrer's periodicity theorem. Moreover, we prove another way to reduce the number of variables in computing the stable category ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ of graded maximal Cohen-Macaulay modules if $A$ is a noncommutative quadric hypersurface. Under high rank property defined in this paper, we also show that computing ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ over a noncommutative smooth quadric hypersurface $A$ in up to six variables can be reduced to one or two variables cases. In addition, we give a complete classification of ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ over a smooth quadric hypersurface $A$ in a skew $\mathbb P^{n-1}$, where $n \leq 6$, without high rank property using graphical methods.

A (2,3)-complete intersection fourfold with no decomposition of the diagonal

We apply the specialization technique based on the decomposition of the diagonal to the intersection of a quadric and cubic hypersurface in mathbb {P}^6. We find an explicit example defined over mathbb {Q} that is smooth, and does not admit a decomposition of the diagonal, and is therefore not retract rational. The proof uses the specialization of Nicaise and Ottem (Tropical degenerations and stable rationality, 2020), who proved that the very general complete intersection of this type is stably irrational using the motivic volume.