We study $F$-signature under proper birational morphisms $\unicode[STIX]{x1D70B}:Y\rightarrow X$, showing that $F$-signature strictly increases for small morphisms or if $K_{Y}\leqslant \unicode[STIX]{x1D70B}^{\ast }K_{X}$. In certain cases, we can even show that the $F$-signature of $Y$ is at least twice as that of $X$. We also provide examples of $F$-signature dropping and Hilbert–Kunz multiplicity increasing under birational maps without these hypotheses.

We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring $(R,\mathfrak{m},k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\mathfrak{m}$-adic topology.

We will develop a theory of multi-pointed non-commutative deformations of a simple collection in an abelian category, and construct relative exceptional objects and relative spherical objects in some cases. This is inspired by a work by Donovan and Wemyss.

Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$ . The Yau algebra, $L(V)$ , is the Lie algebra of derivations of the moduli algebra of $V$ . It is a finite-dimensional solvable algebra and its dimension $\unicode[STIX]{x1D706}(V)$ is the Yau number. Fewnomial singularities are those which can be defined by an $n$ -nomial in $n$ indeterminates. Yau and Zuo [‘A sharp upper estimate conjecture for the Yau number of weighted homogeneous isolated hypersurface singularity’, Pure Appl. Math. Q.12(1) (2016), 165–181] conjectured a bound for the Yau number and proved that this conjecture holds for binomial isolated hypersurface singularities. In this paper, we verify this conjecture for weighted homogeneous fewnomial surface singularities.

We show that the anti-canonical volume of an $n$ -dimensional Kähler–Einstein $\mathbb{Q}$ -Fano variety is bounded from above by certain invariants of the local singularities, namely $\operatorname{lct}^{n}\cdot \operatorname{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.

Li introduced the normalized volume of a valuation due to its relation to K-semistability. He conjectured that over a Kawamata log terminal (klt) singularity there exists a valuation with smallest normalized volume. We prove this conjecture and give an explicit example to show that such a valuation need not be divisorial.

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^{0}(C,E)=r+1$ . For $g\geqslant (2r+2)(2r+1)$ , we show how the bundle $E$ can be recovered from the tangent cone to the generalised theta divisor $\unicode[STIX]{x1D6E9}_{E}$ at ${\mathcal{O}}_{C}$ . We use this to give a constructive proof and a sharpening of Brivio and Verra’s theorem that the theta map $\mathit{SU}_{C}(r){\dashrightarrow}|r\unicode[STIX]{x1D6E9}|$ is generically injective for large values of $g$ .

Let $(R,\mathfrak{m})$ be a Noetherian local ring of characteristic $p>0$. We introduce and study $F$-full and $F$-anti-nilpotent singularities, both are defined in terms of the Frobenius actions on the local cohomology modules of $R$ supported at the maximal ideal. We prove that if $R/(x)$ is $F$-full or $F$-anti-nilpotent for a nonzero divisor $x\in R$, then so is $R$. We use these results to obtain new cases on the deformation of $F$-injectivity.

In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,\mathfrak{m})$ be a local ring; we prove that if $R_{\text{red}}$ is Du Bois, then $H_{\mathfrak{m}}^{i}(R)\rightarrow H_{\mathfrak{m}}^{i}(R_{\text{red}})$ is surjective for every $i$ . We find many applications of this result. For example, we answer a question of Kovács and Schwede [Inversion of adjunction for rational and Du Bois pairs, Algebra Number Theory 10 (2016), 969–1000; MR 3531359] on the Cohen–Macaulay property of Du Bois singularities. We obtain results on the injectivity of $\operatorname{Ext}$ that provide substantial partial answers to questions in Eisenbud et al. [Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), 583–600] in characteristic $0$ . These results can also be viewed as generalizations of the Kodaira vanishing theorem for Cohen–Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen–Macaulayness of the defining ideal of Du Bois singularities, which are characteristic- $0$ analogs and generalizations of results of Singh–Walther and answer some of their questions in Singh and Walther [On the arithmetic rank of certain Segre products, in Commutative algebra and algebraic geometry, Contemporary Mathematics, vol. 390 (American Mathematical Society, Providence, RI, 2005), 147–155]. We extend results on the relation between Koszul cohomology and local cohomology for $F$ -injective and Du Bois singularities first shown in Hochster and Roberts [The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117–172; MR 0417172 (54 #5230)]. We also prove that singularities of dense $F$ -injective type deform.

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy $K$-theory groups of a Noetherian scheme $X$ of Krull dimension $d$ vanish below $-d$. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy $K$-theory groups vanish below $-1$. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy $K$-theory group.

We show that the finiteness of the fundamental groups of the smooth locus of lower dimensional log Fano pairs would imply the finiteness of the local fundamental group of Kawamata log terminal (klt) singularities. As an application, we verify that the local fundamental group of a three-dimensional klt singularity and the fundamental group of the smooth locus of a three-dimensional Fano variety with canonical singularities are always finite.

We give a descent result for formal smoothness having interesting applications: we deduce that quasiexcellence descends along flat local homomorphisms of finite type, we greatly improve Kunz’s characterization of regular local rings by means of the Frobenius homomorphisms as well as André and Radu relativization of this result, etc. In the second part of the paper, we study a similar question for the complete intersection property instead of formal smoothness, giving also some applications.

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 article, we prove that these relations hold only assuming that the algebra is $F$ -pure. In addition, we present an interpretation of the $a$ -invariant for $F$ -pure Gorenstein graded $K$ -algebras in terms of regular sequences that preserve $F$ -purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition $S_{k}$ . We also present analogous results and questions in characteristic zero.

The holomorphy conjecture roughly states that Igusa’s zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article, we prove the holomorphy conjecture for surface singularities that are nondegenerate over $\mathbb{C}$ with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volumes (which appear in the formula of Varchenko for the zeta function of monodromy) of the faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast to the context of the trivial character, we here need to show fakeness of certain candidate poles other than those contributed by $B_{1}$ -facets.

The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, that is, a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is nontrivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a curve of genus 2 illustrating the extent to which these conditions are necessary.

In this article, we consider the conjectured relationship between $F$ -purity and log canonicity for polynomials over $\mathbb{C}$ . In particular, we show that log canonicity corresponds to dense $F$ -pure type for all polynomials whose supporting monomials satisfy a certain nondegeneracy condition. We also show that log canonicity corresponds to dense $F$ -pure type for very general polynomials over $\mathbb{C}$ . Our methods rely on showing that the $F$ -pure and log canonical thresholds agree for infinitely many primes, and we accomplish this by comparing these thresholds with the thresholds associated to their monomial term ideals.

A semialgebraic map $f:X\rightarrow Y$ between two real algebraic sets is called blow-Nash if it can be made Nash (i.e., semialgebraic and real analytic) by composing with finitely many blowings-up with nonsingular centers.

We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ satisfies a lower bound of the Jacobian determinant condition then $f^{-1}$ is also blow-Nash and satisfies the same condition.

The proof relies on motivic integration arguments and on the virtual Poincaré polynomial of McCrory–Parusiński and Fichou. In particular, we need to generalize Denef–Loeser change of variables key lemma to maps that are generically one-to-one and not merely birational.

Let $k$ be field of characteristic zero. Let $f\in k[X,Y]$ be a nonconstant polynomial. We prove that the space of differential (formal) deformations of any formal general solution of the associated ordinary differential equation $f(y^{\prime },y)=0$ is isomorphic to the formal disc $\text{Spf}(k[[Z]])$ .

In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$ . To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity.