Let $\overline{X}$ be a germ of normal surface with local ring $\overline{R}$ covering a germ of regular surface $X$ with local ring $R$ of characteristic $p\>0$. Given an extension of valuation rings $W/V$ birationally dominating $\overline{R}/R$, we study the existence of a new such pair of local rings ${\overline{R}}^{\text{'}}/R^{\text{'}}$ birationally dominating $\overline{R}/R$, such that $R^{\text{'}}$ is regular and ${\overline{R}}^{\text{'}}$ has only toric singularities. This is achieved when $W/V$ is defectless or when $[W:V]$ is equal to $p$

The equivalence of the definitions of the Łojasiewicz exponent introduced by Ha and by Chądzyński and Krasiński is proved. Moreover we show that if the above exponents are less than -1 then they are attained at a curve meromorphic at infinity.

Let $h=\sum h_{\alpha \beta }{X}^{\alpha }{Y}^{\beta }$ be a polynomial with complex coefficients. The Łojasiewicz exponent of the gradient of h at infinity is the least upper bound of the set of all real λ such that ${\left|gradh(x,y)\right|\ge c\left|(x,y)\right|}^{\lambda }$ in a neighbourhood of infinity in ℂ², for c > 0. We estimate this quantity in terms of the Newton diagram of h. Equality is obtained in the nondegenerate case.

In this paper, we show that if $M_1$ and $M_2$ are algebraic real hypersurfaces in (possibly different) complex spaces of dimension at least two and if $f$ is a holomorphic mapping defined near a neighborhood of $M_1$ so that $f(M_1)\subset M_2$, then $f$ is also algebraic. Our proof is based on a careful analysis on the invariant varieties and reduces to the consideration of many cases. After a slight modification, the argument is also used to prove a reflection principle, which allows our main result to be stated for mappings...

Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro’s result on klt-trivial fibrations.

Let $X$ be a projective Frobenius split variety with a fixed Frobenius splitting $\theta$. In this paper we give a sharp uniform bound on the number of subvarieties of $X$ which are compatibly Frobenius split with $\theta$. Similarly, we give a bound on the number of prime $F$-ideals of an $F$-finite $F$-pure local ring. Finally, we also give a bound on the number of log canonical centers of a log canonical pair. This final variant extends a special case of a result of Helmke.

Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$. We consider the image ${𝒩}_1(D,X)$ of ${𝒩}_1\left(D\right)$ in ${𝒩}_1\left(X\right)$ under the natural push-forward of $1$-cycles. We show that ${\rho }_X-{\rho }_D\le codim{𝒩}_1(D,X)\le 8$. Moreover if $codim{𝒩}_1(D,X)\ge 3$, then either $X\cong S\times T$ where $S$ is a Del Pezzo surface, or $codim{𝒩}_1(D,X)=3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that ${\rho }_X-{\rho }_T=4$.