Caucher Birkar
We study certain modules over the algebra of a Cartier divisor on a scheme. Using these modules, we present an inductive method for studying finite generation properties of algebras and modules. In the context of the minimal model program, we show that finite generation of log canonical algebras and modules is equivalent to the minimal model and abundance conjectures.
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We investigate the minimal number of generators $μ$ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound $C$ there exist, up to isomorphism, only finitely divisorial ideals $I$ such that $μ(I)\le C$. It follows that there exist only finitely many Cohen--Macaulay divisor classes. Moreover we determine the minimal depth of all divisorial ideals and the behaviour of $μ$ and depth in ``arithmetic progressions'' in t…
The purpose of this paper is to lay the foundations for the theory of higher rank b-divisorial algebras of Shokurov type. We develop techniques to deal with such objects and propose two natural conjectures regarding Shokurov algebras and adjoint algebras. We confirm these conjectures in the case of affine curves.
Divisorial fans and algebraic torus actions over arbitrary fields
本文通过除数扇和伽罗瓦半线性作用，给出了任意域上正则 $T$-变种的代数几何组合描述。
The Rees algebra and analytic spread of a divisorial filtration
本文研究除子滤链的Rees代数与解析展度，将McAdam定理推广到$\mathbb Q$-除子滤链，证明其一阶差分函数的上极限有限但极限可能不存在，并构造了Rees赋值集合无限的符号滤链例子。
This paper shows a finiteness property of a divisorial valuation in terms of arcs. First we show that every divisorial valuation over an algebraic variety corresponds to an irreducible closed subset of the arc space. Then we define the codimension for this subset and give a formula of the codimension in terms of "relative Mather canonical class". By using this subset, we prove that a divisorial valuation is determined by assigning the values of finite functions. We also have a criterion for a divisorial valuation to be a monomial valuation by assigning the values of finite functions.
Rees Algebras and the reduced fiber cone of divisorial filtrations on two dimensional normal local rings
本研究针对二维正规优局部环上的除子滤过，显式刻画其Rees代数的Proj概型与约化例外纤维结构，给出F. Russo定理的新证明，并明确解析展延不同取值对应的纤维性质。
Using the theory of jet schemes, we give a new approach to the description of a minimal generating sequence of a divisorial valuations on $\textbf{A}^2.$ For this purpose, we show how one can recover the approximate roots of an analytically irreducible plane curve from the equations of its jet schemes. As an application, for a given divisorial valuation $v$ centered at the origin of $\textbf{A}^2,$ we construct an algebraic embedding $\textbf{A}^2\hookrightarrow \textbf{A}^N,N\geq 2$ such that $v$ is the trace of a monomial valuation on $\textbf{A}^N.$ We explain how results in this direction …
Let $V$ be a finite set of divisorial valuations centered at a 2-dimensional regular local ring $R$. In this paper we study its structure by means of the semigroup of values, $S_V$, and the multi-index graded algebra defined by $V$, $\gr_V R$. We prove that $S_V$ is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in $V$, the approximation of a reduced plane curve singularity $C$ by families of sets $V^{(k)}$ o…
Divisorial persistent homology and volume asymptotics of complex pairs
该研究为解析对引入除子渐近同调（DAH），它定义协变函子区分三层渐近信息，建立多种性质，证明正规解析曲面芽的刚性定理，通过例子表明DAH能检测特殊渐近拓扑信息，且高维中除子临界性与同调持续性可能不同。
We consider the last value $\hatμ (ν)$ of the vanishing sequence of $H^0(L)$ along a divisorial or irrational valuation $ν$ centered at $\mathcal{O}_{\mathbb{P}^2,p}$, where $L$ resp. $p$ is a line resp. a point of the projective plane $\mathbb{P}^2$ over an algebraically closed field. This value contains, for valuations, similar information as that given by Seshadri constants for points. It is always true that $\hatμ (ν) \geq \sqrt{1 / \mathrm{vol}(ν)}$ and minimal valuations are those satisfying the equality. In this paper, we prove that the Greuel-Lossen-Shustin Conjecture implies a variati…
We show that the subgraph of the concave transform of a multiplicative filtration on a section ring is the Newton--Okounkov body of a certain semigroup, and if the filtration is induced by a divisorial valuation, then the associated graded algebra is the algebra of sections of a concrete line bundle in higher dimension. We use this description to give a rationality criterion for certain Seshadri constants. Along the way we introduce Newton--Okounkov bodies of abstract graded semigroups and determine conditions for their slices to be Newton--Okounkov bodies of subsemigroups.
Let S be a toric algebra over a field K of characteristic 0 and let I be a monomial ideal of S. We show that the local cohomology modules H^i_I(S) are of finite length over the ring of differential operators D(S;K), generalizing the classical case of a polynomial algebra S. As an application, we compute the characteristic cycles of some local cohomology modules.
We show that for a class of modules over shod algebras, including the canonical tilting modules, the closures of the corresponding orbits in module varieties are regular in codimension one.
We describe the generic modules in each component of the spaces of representations of certain string algebras. In so doing, we calculate the dimensions of higher self-extension groups for generic modules. This algorithm lends itself for use in determining tilting modules over gentle string algebras.
We classify canonical algebras such that for every dimension vector of a regular module the corresponding module variety is normal (respectively, a complete intersection). We also prove that for the dimension vectors of regular modules normality is equivalent to irreducibility.
We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let $V$ be a vertex algebra, $H$ a finite group of automorphisms of $V$, and $C$ an algebraic curve such that $H \subset \on{Aut}(C)$. We show that a suitable collection of twisted $V$--modules gives rise to a section of a certain sheaf on the quotient $X=C/H$. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac-Moody vertex algebras. We also give a chiral …
We propose a notion of algebra of {\it twisted} chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of "smallest" such modules are irreducible $\ghat$-modules and all …
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