In this paper we characterize two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic from the viewpoint of the initial term of the defining equation. As an application, we prove a conjecture about a uniform bound of divisors computing minimal log discrepancies for two dimensional varieties, which is a conjecture by Ishii and also a special case of the conjecture by Mustaţǎ-Nakamura.
In this paper the detailed classification of three-dimensional exceptional canonical hypersurface singularities which don't satisfy the condition of well-formedness is given. This result completes the classification of three-dimensional exceptional log canonical hypersurface singularities started in [4] (math.AG 0201025).
The log canonical thresholds of irreducible quasi-ordinary hypersurface singularities are computed, using an explicit list of pole candidates for the motivic zeta function found by the last two authors.
We prove that if Y is a hypersurface of degree d in P^n with isolated singularities, then the log canonical threshold of (P^n,Y) is at least min{n/d,1}. Moreover, if d is at least n+1, then we have equality if and only if Y is the projective cone over a (smooth) hypersurface in P^{n-1}. In the case when Y is a hyperplane section of a smooth hypersurface in P^{n+1}, Cheltsov and Park have proved that Y has isolated singularities and they have obtained the above lower bound for the log canonical threshold. Moreover they made the conjecture about the equality case (for d=n+1) and they proved that…
We classify four-dimensional quasismooth weighted hypersurfaces with small canonical class, and verify a conjecture of Johnson and Kollar on infinite series of quasismooth hypersurfaces with anticanonical hyperplane section in the case of fourfolds. By considering the quotient singularities that arise, we classify those weighted hypersurfaces that are canonical, Calabi-Yau, and Fano fourfolds. We also consider other classes of hypersurfaces, including Fano hypersurfaces of index greater than 1 in dimensions 3 and 4.
In this article we present a formula for the plurigenera of minimal models of nondegenerate toric hypersurfaces, which is valid in arbitrary dimension and which expresses these invariants through lattice points on the Fine interior. From this formula we derive a formula for the volume of the canonical divisor of the toric hypersurface. We also study the pluricanonical mappings and show under some restrictions how to construct a canonical model of the toric hypersurface.
We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an application, we show that they have a Kaehler-Einstein metric if they are general.
Linearly projecting smooth projective varieties provides a method of obtaining hypersurfaces birational to the original varieties. We show that in low dimension, the resulting hypersurfaces only have Du Bois singularities. Moreover, we conclude that these Du Bois singularities are in fact semi log canonical. However, we demonstrate the existence of counterexamples in high dimension -- the generic linear projection of certain varieties of dimension 30 or higher is neither semi log canonical nor Du Bois.
Let $S$ be a complete intersection of a smooth quadric 3-fold $Q$ and a hypersurface of degree $d$ in ${\mathbb P}^4$. In this paper we analyze GIT stability of $S$ with respect to the natural $G=SO(5, {\mathbb C})$-action. We prove that if $d\ge 4$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-stable. Also, we prove that if $d\ge 3$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-semistable.
We point out an interesting relation between hypersurface elliptic singularities and log Enriques surfaces: with a few exceptions, every hypersurface elliptic singularity define some klt log Enriques surface $(S,Diff)$. In many cases, the log canonical cover of $(S,Diff)$ is a Du Val K3 surface with ``large'' nonsymplectic automorphism group.
We show the birational boundedness of anti-canonical irreducible hypersurfaces which form 3-fold plt pairs. We also treat a collection of Du Val K3 surfaces which is birationally bounded but unbounded.
We establish a link between the GIT stability of complete intersections of same degree hypersurfaces in the same ambient projective space, which can be parametrised as tuples in a Grassmanian scheme, and the log canonical thresholds of hypersurfaces contained in these tuples. We also establish a similar link in the case of Variations of GIT quotients, where we consider tuples of complete intersections and distinct hyperplanes.
We determine, complementing a paper by Marcel Morales, the log-pluricanonical forms on a nondegenerate hypersurface. This description shows that they are extendable to 1-parameter deformations. For an equisingular deformation we thus obtain a simultaneous log-canonical model.
All varieties, extremal contractions, singularities are divided on exceptional and non-exceptional ones. Roughly speaking, there are the infinite families of non-exceptional varieties, extremal contractions or singularities and only the finite number of the types of exceptional ones. This subdivision is well demonstrated by the example of Du Val singularities. There are two infinite series of non-exceptional singularities: $A_n$ and $D_n$ and only three types of exceptional ones: $E_6$, $E_7$ and $E_8$. Also the importance of exceptionality phenomenon follows from the next observation: A). If …
Let $X$ be a smooth hypersurface of degree $n\geq 3$ in $\mathbb{P}^n$. We prove that the log canonical threshold of $H\in|-K_X|$ is at least $\frac{n-1}{n}$. Under the assumption of the Log minimal model program, we also prove that a hyperplane section $H$ of $X$ is a cone in $\mathbb{P}^{n-1}$ over a smooth hypersurface of degree $n$ in $\mathbb{P}^{n-2}$ if and only if the log canonical threshold of $H$ is $\frac{n-1}{n}$.
The purpose of this paper is to point out a relation between the canonical sheaf and the intersection complex of a singular algebraic variety. We focus on the hypersurface case. Let $M$ be a complex manifold, $X\subset M$ a singular hypersurface. We study residues of top-dimensional meromorphic forms with poles along $X$. Applying resolution of singularities sometimes we are able to construct residue classes either in $L^2$-cohomology of $X$ or in the intersection cohomology. The conditions allowing to construct these classes coincide. They can be formulated in terms of the weight filtration. …
We give detailed descriptions of the period maps of two 2-parameter families of anti-canonical hypersurfaces in toric 3-folds. One of them is related to a Hilbert modular surface, and the other is related to the product of modular curves.
For a smooth germ of algebraic variety $(X,0)$ and a hypersurface $(f=0)$ in $X$, with an isolated singularity at $0$, Teissier conjectured a lower bound for the Arnold exponent of $f$ in terms of the Arnold exponent of a hyperplane section $f\vert_H$ and the invariant $θ_0(f)$ of the hypersurface. By building on an approach due to Loeser, we prove the conjecture in the case of log canonical thresholds.
We prove the existence of a canonical zero-cycle on a Calabi-Yau hypersurface X in a complex projective homogeneous variety. More precisely, we show that the intersection of any n divisors on X, n=dim X, is proportional to the class of a point on a rational curve in X.
We prove a precise inversion of adjunction formula for the log pair associated to a non-degenerate hypersurface. As a corollary, the minimal log discrepancies of non-degenerate normal hypersurface singularities are bounded from above by dimension.
Given the same anti-canonical linear system on two distinct toric varieties, we provide a derived equivalence between partial crepant resolutions of the corresponding stacky hypersurfaces. The applications include: a derived unification of toric mirror constructions, calculations of Picard lattices for linear systems of quartic surfaces, and a birational reduction of Reid's list to 81 families.
We investigate the 0-th local cohomology of the Jacobian ring of a homogeneous polynomial defining a projective hypersurface whose singular locus is a 0-dimensional complete intersection.
The main theorem of this paper is that, for a general pair $(A,X)$ of an (ample) Hypersurface $X$ in an Abelian Variety $A$, the canonical map $Φ_X$ of $X$ is birational onto its image if the polarization given by $X$ is not principal (i.e., its Pfaffian $d$ is not equal to $1$). We also show that, setting $g = dim (A)$, and letting $d$ be the Pfaffian of the polarization given by $X$, then if $X$ is smooth and $$Φ_X : X \rightarrow \mathbb{P}^{N:=g+d-2}$$ is an embedding, then necessarily we have the inequality $ d \geq g + 1$, equivalent to $N : = g+d-2 \geq 2 \ dim(X) + 1.$ We also formulat…
Let D = {D_{1},...,D_{l}} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let Ω^{1}_{P^n}(log D) be the logarithmic bundle attached to it. Following [1], we show that Ω^{1}_{P^n}(log D) admits a resolution of lenght 1 which explicitly depends on the degrees and on the equations of D_{1},...,D_{l}. Then we prove a Torelli type theorem when all the D_{i}'s have the same degree d and l >= {{n+d} \choose {d}}+3: indeed, we recover the components of D as unstable smooth hypersurfaces of Ω^{1}_{P^n}(log D). Finally we analyze the cases of one q…
We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study Zariski equisingular families of surface, not necessarily isolated, singularities in $\mathbb{C}^3$. We show that a natural stratification of such a family given by the singular set and the generic family of polar curves provides a Lipschitz stratification in the sense of Mostowski. In particular, such families are bi-Lipschitz trivial by trivializations obtained by integrating Lipschitz vector fields.
We study the Du Bois complex $\underlineΩ_Z^\bullet$ of a hypersurface $Z$ in a smooth complex algebraic variety in terms its minimal exponent $\widetildeα(Z)$. The latter is an invariant of singularities, defined as the negative of the greatest root of the reduced Bernstein-Sato polynomial of $Z$, and refining the log canonical threshold. We show that if $\widetildeα(Z)\geq p+1$, then the canonical morphism $Ω_Z^p\to \underlineΩ_Z^p$ is an isomorphism, where $\underlineΩ_Z^p$ is the $p$-th associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, …
We construct families of quartic and cubic hypersurfaces through a canonical curve, which are parametrized by an open subset in a Grassmannian and a Flag variety respectively. Using G. Kempf's cohomological obstruction theory, we show that these families cut out the canonical curve and that the quartics are birational (via a blowing-up of a linear subspace) to quadric bundles over the projective plane, whose Steinerian curve equals the canonical curve.
On plane algebraic curves the so-called Weierstrass kernel plays the same role of the Cauchy kernel on the complex plane. A straightforward prescription to construct the Weierstrass kernel is known since one century. How can it be extended to the case of more general curves obtained from the intersection of hypersurfaces in a $n$ dimensional complex space? This problem is solved in this work in the case $n=3$. As an application, the correlation functions of bosonic string theories are constructed on a canonical curve of genus four.
Let $ \mathcal{D} = \{D_{1}, \ldots, D_{\ell}\} $ be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space $ \mathbb{P}^{n} $ and let $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ be the logarithmic bundle attached to it. Our aim is to study the injectivity of the correspondence $ \mathcal{D} \longrightarrow Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. In order to do that, we first show that $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ admits a resolution of length $ 1 $ depending on the degrees and on the equations of $ D_{1}, \ldots, D_{\ell} $. Then, we pro…
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.
For a complex algebraic variety $X$, we introduce higher $p$-Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms $Ω_X^q$ and the shifted graded pieces of the Du Bois complex $\underlineΩ_X^q$ for $q\le p$. If $X$ is a reduced hypersurface, we show that higher $p$-Du~Bois singularity coincides with higher $p$-log canonical singularity, generalizing a well-known theorem for $p=0$. The assertion that $p$-log canonicity implies $p$-Du Bois has been proved by Mustata, Olano, Popa, and Witaszek quite recently as a corollary of two theorems assertin…
We show that a generic real projective $n$-dimensional hypersurface of odd degree $d$, such that $4(n-2)=\binom{d+3}3$, contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3\log d$, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.
If $X$ is an algebraic variety with at worst canonical singularities and $S$ is a $\Q$-Cartier hypersurface in $X$, the canonical threshold of the pair $(X,S)$ is the supremum of $c\in\R$ such that the pair $(X,cS)$ is canonical. We show that the set of all possible canonical thresholds of the pairs $(X,S)$, where $X$ is a germ of smooth 3-dimensional variety, satisfies the ascending chain condition. We also deduce a formula for the canonical threshold of $(\C^3,S)$, where S is a Brieskorn singularity.
We prove the Kodaira vanishing theorem for log-canonical and semi-log-canonical pairs. We also give a relative vanishing theorem of Reid--Fukuda type for semi-log-canonical pairs.
In the present paper, we derive an adjunction formula for the Grauert-Riemenschneider canonical sheaf of a singular hypersurface V in a complex manifold M. This adjunction formula is used to study the problem of extending L2-cohomology classes of dbar-closed forms from the singular hypersurface V to the manifold M in the spirit of the Ohsawa-Takegoshi-Manivel extension theorem. We do that by showing that our formulation of the L2-extension problem is invariant under bimeromorphic modifications, so that we can reduce the problem to the smooth case by use of an embedded resolution of V in M. The…
We establish a relative spannedness for log canonical pairs, which is a generalization of the basepoint-freeness for varieties with log-terminal singularities by Andreatta--Wiśniewski. Moreover, we establish a generalization for quasi-log canonical pairs.
From a block-diagonal $(n+1) \times (m+1) \times (m+1)$ tensor symmetric in the last two entries one obtains two varieties: an intersection of symmetric determinantal hypersurfaces $X$ in $n$-dimensional projective space, and an intersection of quadrics $\mathfrak{C}$ in $m$-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of $X$ in terms of $\mathfrak{C}$. We apply our methods to algebraic curves and show how to construct theta characteristics of certain canonical curves of genera 3, 4, and 5, generalizing a classical construction of…
We introduce a diophantine property of a log canonical algebra, and use it to describe the restriction of a log canonical algebra of general type to a log canonical center of codimension one.
We show that any union of slc strata of a Fano log pair with semi-log canonical singularities is simply connected. In particular, Fano log pairs with semi-log canonical singularities are simply connected, which confirms a conjecture of the first author.
We consider the relative Bruce-Roberts number $μ_{BR}^{-}(f,X)$ of a function on an isolated hypersurface singularity $(X,0)$. We show that $μ_{BR}^{-}(f,X)$ is equal to the sum of the Milnor number of the fibre $μ(f^{-1}(0)\cap X,0)$ plus the difference $μ(X,0)-τ(X,0)$ between the Milnor and the Tjurina numbers of $(X,0)$. As an application, we show that the usual Bruce-Roberts number $μ_{BR}(f,X)$ is equal to $μ(f)+μ_{BR}^{-}(f,X)$. We also deduce that the relative logarithmic characteristic variety $LC(X)^-$, obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the co…
For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth threefolds diffeomorphic to connected sums of S^{3} \times S^{3}. In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S^{3} \times S^{3}. This construction is an analogue of that in Friedman [7], Lu and Tian [12] which used only qu…
For a normal subvariety $V$ of ${\bf C}^n$ with a good ${\bf C}^*$-action we give a simple characterization for when it has only log canonical, log terminal or rational singularities. Moreover we are able to give formulas for the plurigenera of isolated singular points of such varieties and of the logarithmic Kodaira dimension of $V\backslash \{0\}$. For this purpose we introduce sheaves of $m$-canonical and $L^{2,m}$-canonical forms on normal complex spaces. For the case of affine varieties with good ${\bf C}^*$-action we give an explicit formula for these sheaves in terms of the grading of t…
We obtain a correct generalization of Shokurov's non-vanishing theorem for log canonical pairs. It implies the base point free theorem for log canonical pairs. We also prove the rationality theorem for log canonical pairs. As a corollary, we obtain the cone theorem for log canonical pairs. We do not need Ambro's theory of quasi-log varieties.
