Daniel J. Hernández
In this note, we derive a formula for the F-pure threshold of diagonal hypersurfaces over a perfect field of prime characteristic. We also calculate the associated test ideal at the F-pure threshold, and give formulas for higher jumping numbers of Fermat hypersurfaces.
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In this paper we find the tight closure of powers of parameter ideals of certain diagonal hypersurface rings. In many cases the associated graded ring with respect to tight closure filtration turns out to be Cohen-Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We determine the tight Hilbert polynomial in the following cases: (1) F-pure diagonal hypersurfaces where number of variables is equal to the degree of defining equation, (2) diagonal hypersurface rings where characteristic of the ring is one less than the degree of defining equation and (3) qu…
In the ring R=K[X,Y,Z]/(X^3+Y^3+Z^3), where K is a field of prime characteristic p other than 3, determining the tight closure of the ideal (X^2, Y^2, Z^2)R had existed as a classic example of the difficulty involved in tight closure computations. We settle this question, compute the Frobenius closure of this ideal, and generalize these results to the diagonal hypersurfaces K[X_1,...,X_n]/(X_1^n + ... + X_n^n).
Let $R=k[x_1, ..., x_n]/(x_1^d + ... + x_n^d)$, where $k$ is a field of characteristic $p$, $p$ does not divide $d$ and $n \geq 3$. We describe a method for computing the test ideal for these diagonal hypersurface rings. This method involves using a characterization of test ideals in Gorenstein rings as well as developing a way to compute tight closures of certain ideals despite the lack of a general algorithm. In addition, we compute examples of test ideals in diagonal hypersurface rings of small characteristic (relative to $d$) including several that are not integrally closed. These examples…
BCM-regularity of diagonal hypersurfaces and plus-pure thresholds in mixed characteristic
本文提出了一种计算plus-pure阈值的新方法，该阈值是log canonical阈值和F-纯阈值的混合特征类比。研究了Fermat型超曲面的BCM-正则性必要条件和充分条件，并建立了混合特征中对角超曲面的plus-pure阈值下界。通过引入Yoshikawa提出的分裂阶序列，给出了混合特征(0,2)中超曲面的plus-pure阈值界，并应用该方法对混合特征(0,2)中的BCM-正则对角超曲面进行了分类。
F-signature functions of diagonal hypersurfaces
研究对角超曲面的F-signature函数，证明其随素数趋于无穷收敛于分段多项式函数，并用于证明极限F-signature和Hilbert-Kunz重数的存在性，对Fermat超曲面给出显式公式并否定回答Watanabe-Yoshida的问题。
On Stability and Denominators of F-pure thresholds in Families of Diagonal Hypersurfaces
本文研究对角超曲面族中 F-pure 阈值的分母行为，提出了两个 $m$ 变量超曲面族，分别展示阈值分母稳定和幂次发散至无穷的特性，并指出该行为取决于素数 $p$ 模最小两个指数的同余类。
The acquisition of the defining equations of Rees algebras is a natural way to study these algebras and allows certain invariants and properties to be deduced. In this paper, we consider Rees algebras of codimension 2 perfect ideals of hypersurface rings and produce a minimal generating set for their defining ideals. Then, using generic Bourbaki ideals, we study Rees algebras of modules with projective dimension one over hypersurface rings. We describe the defining ideal of such algebras and determine Cohen-Macaulayness and other invariants.
We survey our recently proposed method for constructing biholomorphic invariants of quasihomogeneous isolated hypersurface singularities and, more generally, invariants of graded Artinian Gorenstein algebras. The method utilizes certain polynomials associated to such algebras, called nil-polynomials, and we compare them with two other classes of polynomials that have also been used to produce invariants.
Invariants of Non-Isolated Singularities of Hypersurfaces
本文推广了 Siersma、Pellikaan 和 de Jong 关于奇异轨迹为光滑曲线的超曲面 morsification 的结果，通过研究横截判别式及其与代数、拓扑不变量的关系，应用于 Yomdin 型孤立奇点研究。
Limit F-invariants of the simple singularities and the Fermat hypersurfaces
本文计算了简单奇点与费马超曲面的极限F-不变量，推广了相关公式，并将正则环刻画与Watanabe--Yoshida猜想推广至特征零。
The F-pure threshold is a positive characteristic numerical invariant, analogous to the log canonical threshold in characteristic zero. We compute the F-pure threshold of the affine cone over a Calabi-Yau hypersurface, and relate it to the order of vanishing of the Hasse invariant on the versal deformation space of the hypersurface.
On some invariants of hypersurface singularities
研究超曲面奇点的β_P(f)不变量，分析其与log canonical threshold的性质，并探讨最小指数的上界问题。
Diagonal F-thresholds for determinants and Pfaffians
本文研究了行列式和反称超曲面的对角F阈值，通过上同调消失定理和多项式结果，证明了其最小值与a不变量的关系。
Let $A$ be an integral $k$-algebra of finite type over an algebraically closed field $k$ of characteristic $p>0$. Given a collection ${\cal{D}}$ of $k$-derivations on $A$, that we interpret as algebraic vector fields on $X=Spec(A)$, we study the group spanned by the hypersurfaces $V(f)$ of $X$ invariant for ${\cal{D}}$ modulo the rational first integrals of ${\cal{D}}$. We prove that this group is always a finite $\mathbb{Z}/p$-vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant for a fol…
We study singularities f in K[[x_1,...,x_n]] over an algebraically closed field K of arbitrary characteristic with respect to right respectively contact equivalence, and we establish that the finiteness of the Milnor respectively the Tjurina number is equivalent to finite determinacy. We give improved bounds for the degree of determinacy in positive characteristic. Moreover, we consider different non-degeneracy conditions of Kouchnirenko, Wall and Beelen-Pellikaan in positive characteristic, and we show that planar Newton non-degenerate singularities satisfy Milnor's formula mu=2 delta-r+1. Th…
For a prime p>2 and q=p^n, we compute a finite generating set for the SL_2(F_q)-invariants of the second symmetric power representation, showing the invariants are a hypersurface and the field of fractions is a purely transcendental extension of the coefficient field. As an intermediate result, we show the invariants of the Sylow p-subgroups are also hypersurfaces.
Reconstruction of hypersurfaces from their invariants
本文提出一种显式算法，利用共变式理论和 Veronese 嵌入，从一般齐次多项式在 GL_n 或 SL_n 作用下的不变量重建原多项式，并应用于从 Dixmier-Ohno 不变量重建一般非超椭圆亏格 3 曲线及从一般非超椭圆亏格 4 曲线的不变量重建曲线，且重建系数落在模域中。
We study an analytically irreducible algebroid germ (X, 0) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible components of the exceptional divisor of the normalized blow-up of the normalization (Y, 0) of (X, 0), centered at the point 0 of Y. If (X, 0) is a quasi-ordinary hypersurface singularity, we obtain that the associated graded ring is an algebra of finite type over the field of complex numbers, namely the coordinate ring of a non necessarily normal affine tor…
Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases that they considered, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked if differential operators on the corresponding rings of positive prime characteristic lift to characteristic zero differential operators. We prove that, in general, this is not the case for determinantal h…
Let M be a matrix whose entries are power series in several variables and determinant det(M) does not vanish identically. The equation det(M)=0 defines a hypersurface singularity and the (co)-kernel of M is a maximally Cohen-Macaulay module over the local ring of this singularity. Suppose the determinant det(M) is reducible, i.e. the hypersurface is locally reducible. A natural question is whether the matrix is equivalent to a block-diagonal or at least to an upper-block-triangular. (Or whether the corresponding module is decomposable or at least is an extension.) We give various necessary and…
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