John Stalker
Two sets of conditions are presented for the compactness of a real plane algebraic curve, one sufficient and one necessary, in terms of the Newton polygon of the defining polynomial.
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Let $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be a polynomial and $\mathcal{Z}(f)$ its zero set. In this paper, in terms of the so-called Newton polyhedron of $f,$ we present a necessary criterion and a sufficient condition for the compactness of $\mathcal{Z}(f).$ From this we derive necessary and sufficient criteria for the stable compactness of $\mathcal{Z}(f).$
In this paper we discuss some special generalizations of equationally Noetherian property which naturally arise in the universal algebraic geometry. We introduce weakly equationally Noetherian, qw-compact, uw-compact, and weakly uw-compact algebras and then examine properties of such algebras. Also we consider the connections between five classes: the class of equationally Noetherian algebras, the class of weakly equationally Noetherian algebras, the class of uw-compact algebras, the class of weakly uw-compact algebras, and the class of qw-compact algebras.
A criterion for the existence of a plane model with two non-smooth Galois points for algebraic curves is presented, which is a generalization of Fukasawa's criterion for two smooth Galois points. Owing to this generalized criterion, multiplicities and order sequences at Galois points can be described in detail.
In this work we show that the homogeneous space of an affine algebraic group $G$ by a one-dimensional unipotent subgroup $H$ is affine if and only if the subgroup is not contained in any reductive subgroup of $G$.
Gromov's compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov's compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct …
We give an effective criterion for openness of a morphism of schemes of finite type over a field: Over a normal base of dimension n, failure of openness is detected by a vertical component in the n'th fibred power of the morphism. This is a topological analogue of a criterion for flatness that originates with Auslander.
Let f: Y -> CP^2 be a birational morphism of non-singular (rational) surfaces. We give an effective (necessary and sufficient) criterion for algebraicity of the surfaces resulting from contraction of the union of the strict transform of a line on CP^2 and all but one of the exceptional divisors of f. As a by-product we construct normal non-algebraic Moishezon surfaces with the `simplest possible' singularities, which in particular completes the answer to a remark of Grauert. Our criterion involves `global variants' of `key polynomials' introduced by MacLane. The geometric formulation of the cr…
We prove that an algebraic group over a field $k$is affine precisely when its Picard group is torsion, and show that in this case the Picard group is finite when $k$ is perfect, and the product of a finite group of order prime to $p$ and a $p$-primary group of finite exponent when $k$ is imperfect of characteristic $p$.
We give a novel and effective criterion for algebraicity of rational normal analytic surfaces constructed from resolving the singularity of an irreducible curve-germ on $CP^2$ and contracting the strict transform of a given line and all but the `last' of the exceptional divisors. As a by-product we construct a new class of analytic non-algebraic rational normal surfaces which are `very close' to being algebraic. These results are local reformulations of some results in (Mondal, 2011) which sets up a correspondence between normal algebraic compactifications of $C^2$ with one irreducible curve a…
Let $Γ$ be a finitely presented group and $G$ a linear algebraic group over $\mathbb{R}$. A representation $ρ:Γ\rightarrow G(\mathbb{R})$ can be seen as an $\mathbb{R}$-point of the representation variety $\mathfrak{R}(Γ, G)$. It is known from the work of Goldman and Millson that if $Γ$ is the fundamental group of a compact K{ä}hler manifold and $ρ$ has image contained in a compact subgroup then $ρ$ is analytically defined by homogeneous quadratic equations in $\mathfrak{R}(Γ, G)$. When $X$ is a smooth complex algebraic variety, we study a certain criterion under which this same conclusion hol…
We show that in a holomorphic family of compact complex connected manifolds parametrized by an irreducible complex space $S$, assuming that on a dense Zariski open set $S^{*}$ in $S$ the fibres satisfy the $\partial\bar\partial-$lemma, the algebraic dimension of each fibre in this family is at least equal to the minimal algebraic dimension of the fibres in $S^{*}$. For instance, if each fibre in $S^{*}$ are Moishezon, then all fibres are Moishezon.
An affineness criterion for algebraic groups and applications
本文证明光滑连通代数群为仿射群当且仅当任意正规簇上的可逆层均为群不变，并由此给出 Chevalley-Barsotti 结构定理的新简证。
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some topological features of surfaces (reality, compactness, connectivity) and to the problem of plotting.
A criterion for extending morphisms from open subsets of smooth fibrations of algebraic varieties
本文研究代数簇光滑纤维化中开子集上的态射扩张问题，给出一个类似 Zariski 主定理的充分条件，以保证到真态射目标的 $S$-态射可整体扩张。
Fiberwise criteria for Twisted Forms of Algebraic Structures
本文提出了基于闭纤维行为的代数对象形式判据，并以此回答了I. Burban提出的问题。
In this paper, we give a new and efficient algebraic criterion for the pure as well as non-pure shellability of simplicial complex $Δ$ over [n]. We also give an algebraic characterization of a leaf in a simplicial complex (defined in [8]). Moreover, we introduce the concept of Gallai-simplicial complex $Δ_Γ(G)$ of a finite simple graph G. As an application, we show that the face ring of the Gallai simplicial complex associated to tree is Cohen-Macaulay.
Let X be a projective manifold. We prove that the Mabuchi Energy of X is bounded below on all degenerations in B (the space of Bergman metrics) if and only if it is bounded below uniformly on B.
Non-algebraicity of foliations via reduction modulo $2$
本文提出模2约化判据，证明复射影平面上曲线叶层无代数不变曲线，并给出Jouanolou叶层奇数次数情形的新证明及更多例子。
Let $\langle K,ν\rangle$ be a real closed valued field, and let $S\subseteq K^n$ be an open semi-algebraic set. Using tools from model theory, we find an algebraic characterization of rational functions which admit, on $S$, only values in the valuation ring. We use this result to deduce a criterion for a rational function to be bounded on an open semi algebraic subset of some irreducible variety over a real closed field or over an ordered field which is dense in its real closure.
In arXiv:1109.6438v1 [math.AG] we introduced and studied a notion of algebraic entropy. In this paper we will give an application of algebraic entropy in proving Kunz Regularity Criterion for all contracting self-maps of finite length of Noetherian local rings in arbitrary characteristic. Some conditions of Kunz Criterion have already been extended to the general case by Avramov, Iyengar and Miller in arXiv:math/0312412v2 [math.AC], using different methods.
This is the first in a series of two papers concerned with relative birational geometry of algebraic spaces. In this paper, we study Prüfer spaces and Prüfer pairs of algebraic spaces that generalize spectra of Prüfer rings. As a particular case of Prüfer spaces we introduce valuation algebraic spaces, and use them to establish valuative criterion of universal closedness that sharpens the standard criterion. In the sequel paper, we will introduce a version of Riemann-Zariski spaces, and will prove Nagata compactification theorem for algebraic spaces.
Deligne's regularity criterion for an integrable connection $\nabla$ on a smooth complex algebraic variety $X$ says that $\nabla$ is regular along the irreducible divisors at infinity in some fixed normal compactification of $X$ if and only if the restriction of $\nabla$ to every smooth curve on $X$ is regular ({\it i. e.} has only regular singularities at infinity). The ``only if" part is the difficult implication. Deligne's proof is transcendental, and uses Hironaka's resolution of singularities. We give here an elementary and purely algebraic proof of this implication: it is, as far as we k…
We investigate conditions for "simultaneous normalizability" of a family of reduced schemes, i.e., the normalization of the total space normalizes, fiber by fiber, each member of the family. The main result (under more general conditions) is that a flat family of reduced equidimensional projective complex varieties X_y with parameter y ranging over a normal space--algebraic or analytic--admits a simultaneous normalization if and only if the Hilbert polynomial of the integral closure of the structure sheaf O_{X_y} is locally independent of y. When the X_y are curves projectivity is not needed, …
An arithmetic valuative criterion for proper maps of tame algebraic stacks
针对代数栈真态射的通常赋值准则在算术应用中仅给出剩余域扩张上的特化点的问题，本文提出了一种适用于算术应用的驯顺栈真态射新赋值准则，并证明了 Lang-Nishimura 定理对驯顺栈成立。
We show that the Nakai--Moishezon ampleness criterion holds for real line bundles on complete schemes. As applications, we treat the relative Nakai--Moishezon ampleness criterion for real line bundles and the Nakai--Moishezon ampleness criterion for real line bundles on complete algebraic spaces. The main ingredient of this paper is Birkar's characterization of augmented base loci of real divisors on projective schemes.
Deligne's regularity criterion for an integrable connection $\nabla$ on a smooth complex algebraic variety $X$ says that $\nabla$ is regular along the irreducible divisors at infinity in some fixed normal compactification of $X$ if and only if the restriction of $\nabla$ to every smooth curve on $X$ is fuchsian (i.e. has only regular singularities at infinity). The "only if" part is the difficult implication. Deligne's proof is transcendental and uses Hironaka's resolution of singularities. We present a purely algebraic proof of this implication which does not use resolution beyond the case of…
A criterion for the existence of a birational embedding into a projective plane with three collinear Galois points for algebraic curves is presented. The extendability of an automorphism induced by a Galois point to a linear transformation of the projective plane is also discussed, under the assumption that two Galois points exist.
A criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves is presented. A new example of a plane curve with non-collinear Galois points as an application is described. Furthermore, a new characterization of the Fermat curve in terms of non-collinear Galois points is presented.
A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described.
We prove that projectivity is an open condition for deformations of algebraic spaces with rational singularities. V.2: references updated and corrected.
A Criterion for the Algebraic Density Property of Affine $SL_2$-Manifolds
本文针对仿射$SL_2$流形，提出了判断其代数密度性质的代数准则，并将该准则应用于物理学中三类$SL_2$簇的代数密度性质证明。
The tilting correspondence is a fundamental property of perfectoid fields. In this note, we show that the tilting construction can also be used to detect perfectoid fields among nonarchimedean fields. In particular, for $K$ a complete subfield of $\mathbb{C}_p$ (a completed algebraic closure of $\mathbb{Q}_p$), $K$ is perfectoid if and only if its tilt is not algebraic over $\mathbb{F}_p$. We also include some conjectures on APF (arithmetically profinite) extensions, perfectoid fields, and their relations.
A greedy open-orbit criterion for solvable algebraic group actions, with applications to Lusztig's nilpotent varieties
该研究针对可解代数群作用提出贪心开轨道准则，应用于Dynkin箭图无重表示的Aut_Q(M)作用，简化Lusztig幂零簇刚性问题，为等定向A型提供显式算法。
For a tuple $(θ_1,..,θ_M)$ of complex number, buliding on the approximation techniques in earlier papers of this series, this paper engages in deducing lower estimates on the transcendence degree of the field generated by $θ_1, ..., θ_M$ over the field of rational numbers from the approximability of the point $θ=(1,θ_1,...,θ_M)$ in projective space by hypersurfaces. The first given result is an new proof of an algebraic independence criterion, that was formerly proved by Laurent and Roy, and generalizes the Philippon criterion by introducing also evaluations of derivatives of global sections. …
A new criterion for the absolute irreducibility of multivariate polynomials over finite fields
本文提出了一种新的判据，用于判断有限域上由多变量多项式定义的超曲面的绝对不可约性，无需测试基域或扩展域的不可约性，仅需多变量GCD计算和平方自由性质。
Let G be a reductive algebraic group and H a closed subgroup of G. Explicit constructions of G-invariant ideals in the algebra K[G/H] are given. This allows to obtain an elementary proof of Matsushima's criterion: a homogeneous space G/H is an affine variety if and only if H is reductive.
We prove an effective integrability criterion for differential-algebraic Pfaffian systems leading to a decision method of consistency with a triple exponential complexity bound. As a byproduct, we obtain an upper bound for the order of differentiations in the differential Nullstellensatz for these systems.
We present an effective criterion to determine if a normal analytic compactification of C^2 with one irreducible curve at infinity is algebraic or not. As a by product we establish a correspondence between normal algebraic compactifications of C^2 with one irreducible curve at infinity and algebraic curves contained in C^2 with one place at infinity. Using our criterion we construct pairs of homeomorphic normal analytic surfaces with minimally elliptic singularities such that one of the surfaces is algebraic and the other is not. Our main technical tool is the sequence of "key forms" - a 'glob…
We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual t-motives, we find that all algebraic relations among special values of the geometric function field Gamma-function are explained by the standard functional equations.
We address the question of identifying non-smooth points in affine real algebraic varieties. A simple algebraic criterion will be formulated and proven. As an application we can answer several questions about the configuration spaces of some class of planar linkages.
We investigate the smoothing problem of limit linear series of rank one on an enrichment of the notions of nodal curves and metrized complexes called saturated metrized complexes. We give a finitely verifiable full criterion for smoothability of a limit linear series of rank one on saturared metrized complexes, characterize the space of all such smoothings, and extend the criterion to metrized complexes. As applications, we prove that all limit linear series of rank one are smoothable on saturated metrized complexes corresponding to curves of compact-type, and prove an analogue for saturated m…
Let $G$ be a reductive affine algebraic group, and let $X$ be an affine algebraic $G$-variety. We establish a (poly)stability criterion for points $x\in X$ in terms of intrinsically defined closed subgroups $H_{x}$ of $G$, and relate it with the numerical criterion of Mumford, and with Richardson and Bate-Martin-Röhrle criteria, in the case $X=G^{N}$. Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson-Millson and Sikora about complex representation varieties of finitely …
Explicit smooth real algebraic functions which may have both compact and non-compact preimages on smooth real algebraic manifolds
本文提出一种变体，在满足原像紧致性显式条件下于光滑实代数流形上构造非真光滑实代数函数，推进了可微映射奇点理论与实代数几何的发展。
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