Related papers: F-invariants of diagonal hypersurfaces
We investigate the hypersurfaces which are the generation of the Fermat hypersufaces, and determine their projective isomorphism classes.
The a-invariant, the F-pure threshold, and the diagonal F-threshold are three important invariants of a graded K-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly F-regular rings. In this…
We present results expressing conditions for the existence of meromorphic first integrals for Pfaff equations of arbitrary codimension, integrable or not, on complex manifolds. These results are in the same vein as previous ones by J-P.…
We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of…
This paper describes a method for computing all F-pure ideals for a given Cartier map of a polynomial ring over a finite field.
We prove that the variety defined by the determinant of the matrix of diagonals is $F$-pure for matrices of all sizes and in all positive prime characteristics. Moreover, we find a system of parameters for it.
Given an ideal J on a smooth variety in characteristic zero, we estimate the F-jumping numbers of the reductions of J to positive characteristic in terms of the jumping numbers of J and the characteristic. We apply one of our estimates to…
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…
We prove that if $f$ is a reduced homogenous polynomial of degree $d$, then its $F$-pure threshold at the unique homogeneous maximal ideal is at least $\frac{1}{d-1}$. We show, furthermore, that its $F$-pure threshold equals $\frac{1}{d-1}$…
In this paper, we estimate the number of $\mathbb{F}_q$-primitive points on the affine hypersurface defined by the equation $f(x_1,\ldots,x_s)=0$, where $f\in\mathbb{F}_q[x_1,\dots,x_s]$ is an appropriate polynomial. In particular, we…
The generic link of an unmixed radical ideal is radical (in fact, prime). We show that the squarefreeness of the initial ideal and $F$-purity are, however, not preserved along generic links. On the flip side, for several important cases in…
We compute the $F$-pure threshold of the natural cone over flag varieties in characteristic $p>0$. Our calculations are mainly focused on flag varieties that are arithmetically Gorenstein, but we offer some results in the non-Gorenstein…
We study the plus-pure threshold (ppt) of hypersurfaces in mixed characteristic. We show that the ppt limits to the $F$-pure threshold (fpt) as we ramify the base DVR. Additionally, we show that analogs of some positive characteristic…
In this paper, we use big Cohen-Macaulay algebras to define a characteristic free analog of the $F$-thresholds, which we call BCM-thresholds, in the case of principal ideals. We prove that, similarly to the case of the $F$-thresholds, the…
We introduce a new criterion providing a sufficient condition for a hypersurface in an unramified regular local ring to be perfectoid pure. The criterion is formulated in terms of an explicitly computable sequence of integers, called the…
We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…
The purpose of this note is to revisit the results of arXiv:1407.4324 from a slightly different perspective, outlining how, if the integral closures of a finite set of prime ideals abide the expected convexity patterns, then the existence…
We introduce two families of ideals, $F$-jumping ideals and $F$-Jacobian ideals, in order to study the singularities of hypersurfaces in positive characteristic. Both families are defined using the $D$-modules $M_{\alpha}$ that were…
Let X be a smooth hypersurface in projective space. We discuss in this paper when X can be defined by an equation det M = 0 (resp. pf M = 0), where M is a matrix (resp. a skew-symmetric matrix) with homogeneous entries. Standard homological…
We give uniform upper bounds for the number of integral points of bounded height on affine hypersurfaces, which generalise earlier results of Browning,Heath-Brown and the author.