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We prove that the ideal generated by the maximal minors of the higher-order Jacobian matrix of a weighted homogeneous polynomial is also weighted homogeneous. As an application, we give a partial answer to a conjecture concerning the non-existence of negative weight derivations on the higher Nash blowup local algebra of a hypersurface., Comment: Comments are welcome

We begin the study of Lipschitz saturation for germs of toric singularities. By looking at their associated analytic algebras, we prove that if (X,0) is a germ of toric singularity with smooth normalization then its Lipschitz saturation is again toric. Finally we show how to calculate the Lipschitz saturation for some families of toric singularities starting from the semigroup that defines them., Comment: 27 pages, 3 figures

We investigate the relation between essential divisors and F-blowups, in particular, address the problem whether all essential divisors appear on the $e$-th F-blowup for large enough $e$. Focusing on the case of normal affine toric varieties, we establish a simple sufficient condition for a divisor over the given toric variety to appear on the normalized limit F-blowup as a prime divisor. As a corollary, we show that if a normal toric variety has a crepant resolution, then the above problem has a positive answer, provided that we use the notion of essential divisors in the sense of Bouvier and Gonzalez-Sprinberg. We also provide an example of toric threefold singularities for which a non-essential divisor appears on an F-blowup., Comment: 24 pages. v2: minor revision, v3: fixed errors that was pointed out by a referee and based on confusion about several notions of essential divisors, v4: modified the definition of ciritical arrows, added some descriptions to help understanding of the discussion

We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime characteristic version of the logarithmic Jacobian ideal of a toric variety and prove that its blowup coincides with the Nash blowup of the variety. As a consequence, the Nash blowup of a, not necessarily normal, toric variety of arbitrary dimension in prime characteristic can be described combinatorially.

We give an explicit set of generators for the semigroup of the Gr\"obner degeneration of a toric ideal. This set of generators is used to study algebraic properties of the semigroup it generates: approximation of semigroups, non-preservation of saturation, Betti elements, uniqueness of presentations, and M\"obius functions.

We prove that, for the jet scheme of a singular hypersurface, the blowup of a certain jet-related module is not an isomorphism. In conjunction with recent developments in the theory of Nash blowups, our result holds over fields of arbitrary characteristic. Our approach is based on explicit presentations given by a higher-order Jacobian matrix combined with a certain jet-related matrix.

In this letter we present emergent screening of magnetic monopole and dipole by the presence of 20nm aluminum cover layer. Our results were obtained in base of magnetic atomic force measurements, performed after external magnetic field steps application. We show that the evolution of magnetization and monopole population is affected by the aluminum presence and attribute that phenomena to the proximity effect, which is responsible for the magnetization vanish of the firstatomic layers at the interface. Using experimental values to estimate the decrease in the nano-magnetic dipole value used in an emergent excitation model and in the switching field distribution heterogeneity used in simulations, we observe a very good agreement among experimental and simulation results. The presented emergent screening could be used in new ASI geometries for thermodynamic activation or proposition of devices with selective magnetic monopole mobility., Comment: 6 pages

We prove that the higher Nash blowup of a normal toric variety defined over a field of positive characteristic is an isomorphism if and only if it is non-singular. We also extend a result by R. Toh-Yama which shows that higher Nash blowups do not give a one-step resolution of the $A_3$-singularity. These results were previously known only in characteristic zero., Comment: 12 pages, comments welcome

We initiate the study of Nash blowups in prime characteristic. First, we show that a normal variety is non-singular if and only if its Nash blowup is an isomorphism, extending a theorem by A. Nobile. We also study higher Nash blowups, as defined by T. Yasuda. Specifically, we give a characteristic-free proof of a higher version of Nobile's Theorem for quotient varieties and hypersurfaces. We also prove a weaker version for $F$-pure varieties., Comment: 10 pages, comments welcome

In this note we revisit the problem of determining combinatorially the multiplicity at the origin of a toric curve. In addition, we give the exact value of the regularity index of that point for plane toric curves and effective bounds for this number for arbitrary toric curves.

We characterize the $k$-torsion freeness of the module of differentials of order $n$ of a point of a hypersurface in terms of the singular locus of the corresponding local ring.

We revisit the problem of resolution of singularities of toric curves by iterating Nash modification. We give a bound on the number of iterations required to obtain the resolution. We also introduce a different approach on counting iterations by dividing the combinatorial algorithm of Nash modification of toric curves into several division algorithms., Comment: 15 pages

We give an explicit presentation of the module of differentials of order $n$ of a finitely generated algebra via a higher-order Jacobian matrix. We use the presentation to study some aspects of this module in the case of hypersurfaces. More precisely, we prove higher-order versions of known results relating freness and torsion-freness of the module of differentials with the regularity and normality of the hypersurface. We also study its projective dimension.

We introduce a higher-order version of the tangent map of a morphism and find a matrix representation. We then apply this matrix to solve a conjecture by T. Yasuda regarding the semigroup of the higher Nash blowup of formal curves. We first show that the conjecture is true for toric curves. We conclude by exhibiting a family of non-monomial curves where the conjecture fails., Comment: 34 pages

We consider the problem of finding an ideal whose blowup defines the Nash blowup of a toric surface and such that its zero locus coincides with the singular set of the toric surface., Comment: 2 figures

We study the Cohen-Macaulay property of triangular graphs $T_n$. We show that $T_2$, $T_3$ and $T_5$ are Cohen-Macaulay graphs, and that $T_4$, $T_6$, $T_8$ and $T_n$ are not Cohen-Macaulay graphs, for $n\geq 10$. Finally, we prove that over fields of characteristic zero $T_7$ and $T_9$ are Cohen-Macaulay., Comment: 10 pages, 2 figures

The higher Nash blowup of an algebraic variety replaces singular points with limits of certain spaces carrying higher-order data associated to the variety at non-singular points. In this note we will define a higher-order Jacobian matrix that will allow us to make explicit computations concerning the higher Nash blowup of hypersurfaces. Firstly, we will generalize a known method to compute the fiber of this modification. Secondly, we will give an explicit description of the ideal whose blowup gives the higher Nash blowup. As a consequence, we will deduce a higher-order version of Nobile's theorem for normal hypersurfaces., Comment: 21 pages