The degree of the discriminant of irreducible representations

关于判别式为14的特殊三次四折与次数为10的非极小K3曲面的注记

A note on special cubic fourfolds of discriminant 14 and non-minimal K3 surfaces of degree 10

本文研究判别式14的特殊三次四折Hassett除子中的一般三次四折，证明其包含特定的10次非极小K3曲面，明确了该曲面的线构成与二次超曲面包含关系。

关于索引为2、度数为5的Fano三次折面中判别式较低的秩2sheaves

Rank Two Sheaves With Low Discriminant on the Fano Threefold of Index 2 and Degree 5

研究在Fano三次折面X_5上描述判别式较低的秩2sheaves的稳定性条件及最大陈类特性。

A relation between number of integral points, volumes of faces and degree of the discriminant of smooth lattice polytopes

We present a formula for the degree of the discriminant of a smooth projective toric variety associated to a lattice polytope P, in terms of the number of integral points in the interior of dilates of faces of dimension greater or equal than $\lceil \frac {\dim P} 2 \rceil$.

Matrix Factorizations of the discriminant of $S_n$

Consider the symmetric group $S_n$ acting as a reflection group on the polynomial ring $k[x_1, \ldots, x_n]$, where $k$ is a field such that Char$(k)$ does not divide $n!$. We use Higher Specht polynomials to construct matrix factorizations of the discriminant of this group action: these matrix factorizations are indexed by partitions of $n$ and respect the decomposition of the coinvariant algebra into isotypical components. The maximal Cohen-Macaulay modules associated to these matrix factorizations give rise to a noncommutative resolution of the discriminant and they correspond to the nontri…

退化判别式

Degenerating Discriminants

研究射影对偶簇和判别式在平坦退化下的行为，通过格罗贝纳退化等方法，描述退化特殊纤维情况并推广到格拉斯曼流形，还得到相关经典公式并联系混合判别式，展现对偶簇和判别式退化行为及应用。

Discriminant loci of ample and spanned line bundles

Let $(X,L,V)$ be a triplet where $X$ is an irreducible smooth complex projective variety, $L$ is an ample and spanned line bundle on $X$ and $V\subseteq H^0(X,L)$ spans $L$. The discriminant locus $\Cal D(X,V) \subset |V|$ is the algebraic subset of singular elements of $|V|$. We study the components of $\Cal D(X,V)$ in connection with the jumping sets of $(X,V)$, generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of $(X,L,V)$) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for t…

关于判别次数为八的锥面的不可理化性

On the unirationality of conic bundles with discriminant of degree eight

本文研究了判别次数为八的锥面的不可理化性，通过构造有理多截面和变形理论分析，揭示了其在不同域上的不可理化性质。

Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]

We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the valuation of the discriminant, assuming the input polynomial F square-free and K a perfect field of characteristic zero or greater than deg(F). The algorithm uses the theory of approximate roots and may be seen as a generalization of Abhyankhar's irreducibility criterion to the case of non algebraically closed residue fields. More generally, we show that we can test within the same complexity if a polynomial is pseudo-irreducible, a larger class of polynomials containing irreducible one…

Discriminant of symmetric matrices as a sum of squares and the orthogonal group

It is proved that the discriminant of $n\times n$ real symmetric matrices can be written as a sum of squares, where the number of summands equals the dimension of the space of $n$-variable spherical harmonics of degree $n$. The representation theory of the orthogonal group is applied to express the discriminant of three by three real symmetric matrices as a sum of five squares, and to show that it can not be written as the sum of less than five squares. It is proved that the discriminant of four by four real symmetric matrices can be written as a sum of seven squares. These improve results of …

判别式24的三次四重簇与有理性质

Cubic fourfolds of discriminant 24 and rationality

本文研究判别式24的三次四重簇，通过其扭曲曲面中Brauer类消失的条件，证明了有理性质，并构造了可数无穷多个新的有理例子。

节点曲面的簇、编码理论与三次超曲面的判别式。第一部分：一般理论与节点K3曲面。第二部分：三次超曲面及其关联判别式。第三部分：节点五次曲面。第四部分：节点六次曲面

Varieties of Nodal surfaces, coding theory and Discriminants of cubic hypersurfaces. Part 1: Generalities and nodal K3 surfaces. Part 2: Cubic Hypersurfaces, associated discriminants. Part 3: Nodal quintics. Part 4: Nodal sextics

本文通过为射影节点曲面附加二元码，系统研究了P^3中节点曲面的Severi簇的不可约分量及其关联层级，完整分类了节点K3曲面，并揭示了Togliatti五次曲面及最大节点数六次曲面与三次超曲面判别式之间的深刻联系。

Discriminants of Symmetric Polynomials

A homogeneous polynomial S(x_1, ..., x_n) of degree r in n variables posesses a discriminant D_{n|r}(S), which vanishes if and only if the system of equations dS/dx_i = 0 has non-trivial solutions. We give an explicit formula for discriminants of symmetric (under permutations of x_1, ..., x_n) homogeneous polynomials of degree r in n >= r variables. This formula is division free and quite effective from the computational point of view: symbolic computer calculations with the help of this formula take seconds even for n ~ 20. We work out in detail the cases r = 2,3,4 which will be probably impo…

The tropical discriminant in positive characteristic

We study singularities in tropical hypersurfaces defined by a valuation over a field of positive characteristic. We provide a method to compute the set of singular points of a tropical hypersurface in positive characteristic and the p-adic case. This computation is applied to determine all maximal cones of the tropical linear space of univariate polynomials of degree $n$ and characteristic $p$ with a fixed double root and the fan of all tropical polynomials that have $0$ as a double root independently of the characteristic. We also compute, by pure tropical means, the number of vertices, edges…

A McKay correspondence for reflection groups

We construct a noncommutative desingularization of the discriminant of a finite reflection group $G$ as a quotient of the skew group ring $A=S*G$. If $G$ is generated by order two reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement $\mathcal{A}(G)$ viewed as a module over the coordinate ring $S^G/(Δ)$ of the discriminant of $G$. This yields, in particular, a correspondence between the nontrivial irreducible representations of $G$ to certain maximal Cohen--Macaulay modules over the coordinate ring $S^G/(Δ)$. These maximal Cohen--Macaulay modules a…

关于双次数 $(m, n)$ 超曲面的判别轨迹的注记

Note sur le lieu discriminant d'une hypersurface de bi-degré $(m, n).$

本文提出一种基于群胚描述单值性的新拓扑方法，用于研究射影空间乘积中代数簇的判别轨迹，并以双次数 $(m, n)$ 超曲面为例进行说明。

Division theorems for the rational cohomology of discriminant complements and applications to automorphism groups in finite characteristic

In this note we extend some of the results of a previous paper \url{arXiv:math/0511593} to algebraically closed fields of finite characteristic. In particular, we show that there is an explicit expression in $n$ and $d$ which is divisible by the prime to $p$ part of the order of the the automorphism group of a smooth degree $d$ hypersurface $\subset \mathbb{P}^n_k$ for $k$ an algebraically closed field of characteristic $p$.

判别式补空间 motive 的误差项与一个 Cayley-Bacharach 定理

Error terms for the motives of discriminant complements and a Cayley-Bacharach theorem

本文在简化假设下证明了关于Cayley-Bacharach集的相关猜想，推导其位于低次曲线的结论，并以此估计光滑超曲面空间归一化motive的误差项，同时证明高维下motive的收敛性等判别式补空间相关结果。

The 3-cuspidal quartic and braid monodromy of degree 4 coverings

We study the discriminant of a degree 4 extension given by a deformed bidouble cover, i.e., by equations z^2= u + a w, w^2= v + bz. We first show that the discriminant surface is a quartic which is cuspidal on a twisted cubic, i.e.,is the discriminant of the general equation of degree 3. We then take a(u,v), b(u,v) and get a 3-cuspidal affine quartic curve whose braid monodromy we compute. This calculation of the local braid monodromy is a step towards the determination of global braid monodromies, e.g. for the (a,b,c) surfaces previously considered by the authors. In the revision we fill a ga…

Discriminant Complements and Kernels of Monodromy Representations

We show that the kernel of the monodromy representation for hypersurfaces of degree d and dimension n is large for d at least three with the exception of the cases (d,n) = (3,0) and (3,1). For these the kernel is finite. By "large" we mean a group that admits a homomorphism to a semisimple Lie group of noncompact type with Zariski-dense image. By the Tits alternative a large group contains a free subgroup of rank two.

Fast construction of irreducible polynomials over finite fields

We present a randomized algorithm that on input a finite field $K$ with $q$ elements and a positive integer $d$ outputs a degree $d$ irreducible polynomial in $K[x]$. The running time is $d^{1+ε(d)} \times (\log q)^{5+ε(q)}$ elementary operations. The function $ε$ in this expression is a real positive function belonging to the class $o(1)$, especially, the complexity is quasi-linear in the degree $d$. Once given such an irreducible polynomial of degree $d$, we can compute random irreducible polynomials of degree $d$ at the expense of $d^{1+ε(d)} \times (\log q)^{1+ε(q)}$ elementary operations …

Irreducible components of the eigencurve of finite degree are finite over the weight space

Let p be a rational prime and N a positive integer which is prime to p. Let W be the p-adic weight space for GL_{2,Q}. Let C_N be the p-adic Coleman-Mazur eigencurve of tame level N. In this paper, we prove that any irreducible component of C_N which is of finite degree over W is in fact finite over W. Combined with an argument of Chenevier and a conjecture of Coleman-Mazur-Buzzard-Kilford (which has been proven in special cases, and for general quaternionic eigencurves) this shows that the only finite degree components of the eigencurve are the ordinary components.

A new determinantal formula for the classical discriminant

According to several classical results by Bezout, Sylvester, Cayley, and others, the classical discriminant D_n of degree n polynomials may be expressed as the determinant of a matrix whose entries are much simpler polynomials in the coefficients of f. However, all of the determinantal formulae for D_n appearing in the classical literature are equivalent in the sense that the cokernels of their associated matrices are isomorphic as modules over the associated polynomial ring. This begs the question of whether there exist formulae which are not equivalent to the classical formulae and not trivi…

Generating series for irreducible polynomials over finite fields

We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the case of the multi-degree and the case of indecomposable polynomials.

An explicit description of the irreducible components of the set of matrix pencils with bounded normal rank

The set of mxn singular matrix pencils with normal rank at most r is an algebraic set with r+1 irreducible components. These components are the closure of the orbits (under strict equivalence) of r+1 matrix pencils which are in Kronecker canonical form. In this paper, we provide a new explicit description of each of these irreducible components which is a parametrization of each component. Therefore one can explicitly construct any pencil in each of these components. The new description of each of these irreducible components consists of the sum of r rank-1 matrix pencils, namely, a column pol…