Trond Gustavsen, Dan Laksov, Roy Skjelnes
We give a short, elementary and explicit proof of the existence of Hilbert schemes of points on affine schemes. As a direct consequence we obtain the existence of the Hilbert scheme of points on any projective scheme, not necessarily of finite type, over any base scheme.
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We generalize the Bialynicki-Birula decomposition to singular schemes and apply it to the Hilbert scheme of points on an affine space. We find an infinite family of small, elementary and generically smooth components of the Hilbert scheme of points of the affine four-space. Our method gives easily verifiable sufficient conditions for proving that a point of the Hilbert scheme is smooth and lies on an elementary component. We also present a necessary condition for smoothability of a finite subscheme given by a homogeneous ideal.
New elementary components of the Gorenstein locus of the Hilbert scheme of points
本研究构造了希尔伯特函数为$(1,n,n,1)$的不可光滑化Gorenstein代数新显式例子，得到点集希尔伯特概型Gorenstein轨迹的一类新无限基本分支，解决了Iarrobino猜想的三次情形。
We answer an open problem posed by Iarrobino in the '80s: is there an elementary component of the Hilbert scheme of points $\textrm{Hilb}^d(\mathbb{A}^n)$ with dimension less than $(n-1)(d-1)$? We construct an infinite class of such components in $\textrm{Hilb}^d(\mathbb{A}^4)$. Our techniques also allow us to construct an explicit example of a local Artinian ring with trivial negative tangents, vanishing nonnegative obstruction space, and socle-dimension $2$.
Galois closures and elementary components of Hilbert schemes of points
该研究推广了有限秩环扩张的函子性Galois闭包，构造了点Hilbert概形的两类新的无限初等不可约分支族，一类参数化单点支撑代数，另一类由Galois闭包模基座元素得到。
The criterion for an affine primary algebra over the field to be integral, is proven. Using this criterion we give a simple proof that Hilbert scheme of 0-dimensional subschemes of length $l$ of nonsingular $d$-dimensional algebraic variety is reduced for all $d$ and $l$.
New components of Hilbert schemes of points and 2-step ideals
本文通过引入并分析2步理想的形变，构造了点Hilbert概形及其嵌套变体的新不可约分支，证明了若干可约性结果，并显式给出215个新的一般约化初等分支。
We prove elementary recursive bounds in the degrees for Positivstellensatz and Hilbert 17-th problem, which is the expression of a nonnegative polynomial as a sum of squares of rational functions. We obtain a tower of five exponentials. A precise bound in terms of the number and degree of the polynomials and their number of variables is provided in the paper.
Components of the nested Hilbert scheme of few points
本文研究了有限点嵌套希尔伯特方案的组成部分，提出了一种系统方法来构建非约简组件，并展示了具有高多重奇点的希尔伯特方案的不可约性。
An elementary proof of Hilbert's theorem on ternary quartics: Some complements
本文在2012年提出的初等证明基础上，进一步简化了Hilbert关于实系数非负三元四次型可表示为三个二次型平方和的定理证明。
In 1888, Hilbert proved that every non-negative quartic form f=f(x,y,z) with real coefficients is a sum of three squares of quadratic forms. His proof was ahead of its time and used advanced methods from topology and algebraic geometry. Up to now, no elementary proof is known. Here we present a completely new approach. Although our proof is not easy, it uses only elementary techniques. As a by-product, it gives information on the number of representations f=p_1^2+p_2^2+p_3^2 of f up to orthogonal equivalence. We show that this number is 8 for generically chosen f, and that it is 4 when f is ch…
Let $K$ be an algebraically closed field of characteristic 0, and let $H$ denote the Hilbert scheme of $m$ points of affine n-space $A^n$. An elementary component $E$ of $H$ is an irreducible component such that every $K$-point $[I]$ in $E$ represents a length-$m$ closed subscheme Spec$(K[x_1,\dots,x_n]/I)$ of $A^n$ that is supported at one point. Iarrobino and Emsalem gave the first explicit examples (with $m > 1$) of elementary components in ["Some zero-dimensional generic singularities: Finite algebras having small tangent space", Comp. Math. 36 (1978), pp. 145-188]; in their examples, the …
We investigate some aspects of the geometry of two classical generalisations of the Hilbert schemes of points. Precisely, we show that parity conjecture for $\text{Quot}_r^d\mathbb{A}^3$ already fails for $d=8$ and $r=2$ and that lots of the elementary components of the nested Hilbert schemes of points on smooth quasi-projective varieties of dimension at least 4 are generically non-reduced. We also deduce that nested Hilbert schemes of points on smooth surfaces have generically non-reduced components. Finally, we give an infinite family of elementary components of the classical Hilbert schemes…
We give an elementary proof of Grothendieck's non-vanishing Theorem: For a finitely generated non-zero module $M$ over a Noetherian local ring $A$ with maximal ideal $\m$, the local cohomology module $H^{\dim M}_{\m}(M)$ is non-zero.
We present a new and direct proof of Grothendieck's generic freeness lemma in its general form. Unlike the previously published proofs, it does not proceed in a series of reduction steps and is fully constructive, not using the axiom of choice or even the law of excluded middle. It was found by unwinding the result of a general topos-theoretic technique.
Let $K$ be an algebraically closed field of characteristic $0$, and let $H^μ$ denote the Hilbert scheme of $μ$ points of the affine space $A^n$. An elementary component $E$ of $H^μ$ is an irreducible component such that every $K$-point $[I]$ $\in$ $E$ represents a length-$μ$ closed subscheme Spec$(K[x_1,\dots,x_n]/I)$ $\subseteq$ $A^n$ that is supported at one point. In a previous article we found some new examples of elementary components; in this article, we simplify the methods and extend the range of the previous paper to find several more examples. In addition, we present a "plausibility …
We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let $A$ be an $n \times n$ symmetric matrix with entries in the polynomial ring $\mathbb R[x_1,...,x_m]$. The result is that if $A$ is postive semidefinite for all substitutions $(x_1,...,x_m) \in \mathbb R^m$, then $A$ can be expressed as a sum of squares of symmetric matrices with entries in $\mathbb R(x_1,...,x_m)$. Moreover, our proof is constructive and gives explicit representations modulo the scalar case.
We give a short and elementary proof of Jung's theorem, which states that for a field K of characteristic zero the automorphisms of K[x,y] are generated by elementary automorphisms and linear automorphisms.
We give an elementary proof of the real section conjecture for quasi-projective hyperbolic curves and semi-abelian varieties. The underlying argument is essentially equivalent to the one given by J. Stix.
An elementary proof of Cohen-Gabber theorem in the equal characteristic $p>0$ case
本文提供了一个新的初等证明，用于Cohen-Gabber定理在等特征$p>0$情况下的应用。
This paper gives a new elementary proof of the theorem that all vector bundles on $\mathbb P^1$ split into the direct sum of line bundles. The proof is based on the study of divisors associated to germs of sections at the generic point.
We give an elementary proof of the group law for elliptic curves using explicit formulas.
This paper proposes a totally constructive approach for the proof of Hilbert's theorem on ternary quartic forms. The main contribution is the ladder technique, with which the Hilbert's theorem is proved vividly.
We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation L such that L(g) is a real rational function. Then we interpret the result in terms of Fuchsian differential equations whose general solution is a polynomial and in terms of electrostatics.
The group structure on the rational points of elliptic curves plays several important roles, in mathematics and recently also in other areas such as cryptography. However, the famous proofs for the group property (in particular, for its associative law) require somewhat advanced mathematics and therefore are not easily accessible by non-mathematician. On the other hand, there have been attempts in the literature to give an elementary proof, but those rely on computer-aided calculation for some part in their proofs. In this paper, we give a self-contained proof of the associative law for this o…
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