Lawrence Ein, Mircea Mustaţǎ
In this note we build on our previous work with Takehiko Yasuda to prove a precise version of inversion of adjunction for varieties which are local complete intersections.
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On $F$-pure inversion of adjunction
本文在正特征情形下研究除子对 $F$-纯性的伴随反演，在 $\mathbb{Q}$-Gorenstein 假设下完整解决主除子情形，并刻画了 $(R, Δ+ D)$ 与 $(R_D, \text{Diff}_D(Δ))$ 的 $F$-纯性关系。
We first announce our recent result on adjunction and inversion of adjunction. Then we clarify the relationship between our inversion of adjunction and Hacon's inversion of adjunction for log canonical centers of arbitrary codimension.
We establish adjunction and inversion of adjunction for log canonical centers of arbitrary codimension in full generality.
In this article we give two independent proofs of the positive characteristic analog of the log terminal inversion of adjunction. We show that for a pair $(X, S+B)$ in characteristic $p>0$, if $(S^n, B_{S^n})$ is strongly $F$-regular, then $S$ is normal and $(X, S+B)$ is purely $F$-regular near $S$. We also answer affirmatively an open question about the equality of $F$-Different and Different.
Inversion of adjunction for higher rational singularities
本文证明了高阶有理奇点的邻接反转。
We prove inversion of adjunction on log canonicity.
We prove several results about the behavior Du Bois singularities and Du Bois pairs in families. Some of these generalize existing statements about Du Bois singularities to the pair setting while others are new even in the non-pair setting. We also prove a new inversion of adjunction result for Du Bois and rational pairs. In the non-pair setting this asserts that if a family over a smooth base has a special fiber $X_0$ with Du Bois singularities and the general fiber has rational singularities, then the total space has rational singularities near $X_0$.
We generalize the notions of F-regular and F-pure rings to pairs $(R,\a^t)$ of rings $R$ and ideals $\a \subset R$ with real exponent $t > 0$, and investigate these properties. These ``F-singularities of pairs'' correspond to singularities of pairs of arbitrary codimension in birational geometry. Via this correspondence, we prove Inversion of Adjunction of arbitrary codimension, which states that for a pair $(X,Y)$ of a smooth variety $X$ and a closed subscheme $Y \subsetneq X$, if the restriction $(Z, Y|_Z)$ to a normal $\Q$-Gorenstein closed subvariety $Z \subsetneq X$ is klt (resp. lc), the…
Inversion of Adjunction for the minimal exponent
本文证明了局部完全交的最小指数满足伴随反转性质，并由此推导出高阶杜布瓦奇点与高阶有理奇点的伴随反转结论。
We prove a result on the inversion of adjunction for log canonical pairs that generalizes Kawakita's result to log canonical centers of arbitrary codimension.
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.
Log canonical inversion of adjunction
本文是一篇简短注记，研究伴随的对数典范反演问题。
Let $(R,Δ+D)$ be a log $\mathbb{Q}$-Gorenstein pair where $R$ is a Noetherian, $F$-finite, normal, local domain of characteristic $p > 0$, $Δ$ is an effective $\mathbb{Q}$-divisor and $D$ is an integral $\mathbb{Q}$-Cartier divisor. We show that the left derivative of the $F$-signature function $s(R,Δ+ tD)$ at $t = 1$ is equal to $-s(\mathcal{O}_D, \mathrm{Diff}_D(Δ))$. This equality is interpreted as a quantitative form of inversion of adjunction for strong $F$-regularity. As an immediate corollary, we obtain the inequality $s(R,Δ) \geq s(\mathcal{O}_D, \mathrm{Diff}_D(Δ))$. We also discuss t…
Inversion of adjunction for quotient singularities
本文研究商奇点中的伴随反转问题，建立 klt Cartier 除子的精确公式，并据此证明 klt 超商奇点最小对数偏差的半连续性。
Inversion of adjunction for quotient singularities II: Non-linear actions
该研究证明了商奇点的精确伴随反演公式，推导了超商奇点极小对数偏差的半连续性，作为此前研究的续篇将相关结果推广至非线性群作用场景。
Inversion of adjunction for quotient singularities III: semi-invariant case
本文证明了有限线性群商的完全交集变种中半不变方程的反向 adjunction 公式，并证明了这些变种的最小 log 可辨性半连续性，扩展了前文关于不变方程的结果。
Let $X$ be a smooth projective $n$-fold such that $q(X)=0$ and $L$ a globally generated, big line bundle on $X$ such that $h^0(K_X+(n-2)L) >0$. We give necessary and sufficient conditions for the adjoint systems $|K_X+kL|$ to be birational for $k \geq n-1$. In particular, for Calabi-Yau $n$-folds we generalize and prove parts of a conjecture of Gallego and Purnaprajna.
We use the theory of motivic integration for singular spaces to give a characterization of minimal log discrepencies in terms of the codimension of certain subsets of spaces of arcs. This is done for arbitrary pairs $(X,Y)$, with $X$ normal and Q-Gorenstein. As a first application, we prove a precise version of Inversion of Adjunction, in the case when the ambient variety is smooth. Another application concerns the semicontinuity of minimal log discrepancies on smooth varieties.
Let $X$ be a complex projective manifold, $L$ an ample line bundle on $X$, and assume that we have a $\mathbb{C}^*$ action on $(X,L)$. We classify such triples $(X,L,\mathbb{C}^*)$ for which the closure of a general orbit of the $\mathbb{C}^*$ action is of degree $\leq 3$ with respect to $L$ and, in addition, the source and the sink of the action are isolated fixed points, and the $\mathbb{C}^*$ action on the normal bundle of every fixed point component has weights $\pm 1$. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano …
Foliation adjunction
本文建立代数簇上叶状结构的伴随公式，并将其用于推进秩一叶状结构的锥定理及奇点研究。
In this paper we give a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we define two convex-geometric notions: the Q-codegree and the nef value of a rational polytope P. We define the adjoint polytope P^(s) as the set of those points in P, whose lattice distance to every facet of P is at least s. We prove a structure theorem for lattice polytopes P with high Q-codegree. If P^(s) is empty for some s < 2/(dim(P)+2), then the lattice polytope P has lattice width one. This has consequences in Ehrhart theory a…
We prove the existence of Fano ladders on weak log Fano varieties of coindex less than 4, as an application of adjunction and nonvanishing.
On the Canonical Bundle Formula and Adjunction for Generalized Kaehler Pairs
该研究针对Kaehler簇上的广义对，证明了典范丛公式、次伴随及伴随逆定理的类比结果，推导了BDPP猜想与锥定理的关联，并完善了已有相关结论。
In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.
Complex projective algebraic varieties with $\mathbb{C}^*$-actions can be thought of as geometric counterparts of birational transformations. In this paper we describe geometrically the birational transformations associated to rational homogeneous varieties endowed with a $\mathbb{C}^*$-action with no proper isotropy subgroups.
We prove a part of Shokurov's conjecture on characterization of toric varieties modulo the minimal model program and adjunction conjecture.
Using inversion of adjunction, we deduce from Nadel's theorem a vanishing property for ideals sheaves on projective varieties, a special case of which recovers a result due to Bertram--Ein--Lazarsfeld. This enables us to generalize to a large class of projective schemes certain bounds on Castelnuovo--Mumford regularity previously obtained by Bertram--Ein--Lazarsfeld in the smooth case and by Chardin--Ulrich for locally complete intersection varieties with rational singularities. Our results are tested on several examples.
We construct families of non-toric $\mathbb{Q}$-factorial terminal Fano ($\mathbb{Q}$-Fano) threefolds of codimension $\geq 20$ corresponding to 54 mutation classes of rigid maximally mutable Laurent polynomials. From the point of view of mirror symmetry, they are the highest codimension (non-toric) $\mathbb{Q}$-Fano varieties for which we can currently establish the Fano/Landau-Ginzburg correspondence. We construct 46 additional $\mathbb{Q}$-Fano threefolds with codimensions of new examples ranging between 19 and 10. Some of these varieties will be presented as toric complete intersections, a…
We show that an element $w$ of a finite Weyl group $W$ is rationally smooth if and only if the hyperplane arrangement $I$ associated to the inversion set of $w$ is inductively free, and the product $(d_1+1) \cdots (d_l+1)$ of the coexponents $d_1,\ldots,d_l$ is equal to the size of the Bruhat interval $[e,w]$, where $e$ is the identity in $W$. As part of the proof, we describe exactly when a rationally smooth element in a finite Weyl group has a chain Billey-Postnikov decomposition. For finite Coxeter groups, we show that chain Billey-Postnikov decompositions are connected with certain modular…
We show that the fundamental class in K-homology of a Frobenius split scheme can be computed as a certain alternating sum over irreducible varieties, with the coefficients computed using Möbius inversion on a certain poset. If G/P is a generalized flag manifold and X is an irreducible subvariety homologous to a multiplicity-free union of Schubert varieties, then using a result of Brion we show how to compute the K_0-class [X] in K_0(G/P) from the Chow class in A_*(G/P).
These notes are based on three lectures given at the 2013 CIME/CIRM summer school. The purpose of this series of lectures is to introduce the notion of a toric fibration and to give its geometrical and combinatorial characterizations. Polarized toric varieties which are birationally equivalent to projective toric bundles are associated to a class of polytopes called Cayley polytopes. Their geometry and combinatorics have a fruitful interplay leading to fundamental insight in both directions. These notes will illustrate geometrical phenomena, in algebraic geometry and neighboring fields, which …
In this article we study the deformations of hyperelliptic polarized varieties $(X,L)$ of dimension $m$ and sectional genus $g$ such that the image $Y$ of the morphism $φ$ induced by $|L|$ is smooth. If $L^m < 2g-2$, it is known that, by adjunction and the Clifford's theorem, any deformation of $(X,L)$ is hyperelliptic. Thus, we focus on when $L^m=2g-2$ or $L^m=2g$. We prove that, if $(X,L)$ is Fano-K3, then, except when $Y$ is a hyperquadric, all deformations of $(X,L)$ are again hyperelliptic (if $Y$ is a hyperquadric, the general deformation of $φ$ is an embedding). This contrasts with the …
A counterexample to the PIA conjecture for minimal log discrepancies
本文通过构造反例，推翻了关于最小 log 抛物线差异的 PIA 猜想，并对家庭中的 LSC 猜想也提供了反例。
We describe a practical and effective method for reconstructing the deformation class of a Fano manifold X from a Laurent polynomial f that corresponds to X under Mirror Symmetry. We explore connections to nef partitions, the smoothing of singular toric varieties, and the construction of embeddings of one (possibly-singular) toric variety in another. In particular, we construct degenerations from Fano manifolds to singular toric varieties; in the toric complete intersection case, these degenerations were constructed previously by Doran--Harder. We use our method to find models of orbifold del …
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