Adjoint rings are finitely generated

On unramified finitely generaed extensions of polynomial rings over a field

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Some non-finitely generated Cox rings

We give a large family of weighted projective planes, blown up at a smooth point, that do not have finitely generated Cox rings. We then use the method of Castravet and Tevelev to prove that the moduli space of stable n-pointed genus zero curves does not have a finitely generated Cox ring if n is at least 13.

Equivariant Cox ring

We define the equivariant Cox ring of a normal variety with algebraic group action. We study algebraic and geometric aspects of this object and show how it is related to the ordinary Cox ring. Then, we specialize to the case of normal rational varieties of complexity one under the action of a connected reductive group G. We show that the G-equivariant Cox ring is then a finitely generated integral normal G-algebra. Under a mild additional condition, we give a presentation by generators and relations of its subalgebra of U-invariants, where U is the unipotent part of a Borel subgroup of G. The …

Examples of non-finitely generated Cox rings

We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of previous work where toric surfaces of Picard number 1 were studied. In this article we consider toric varieties of higher Picard number and higher dimension. In particular, we bring examples of weighted projective 3-spaces blown up at a point that do not have finitely generated Cox rings.

Adjoint ideals and Gorenstein blowups in two-dimensional regular local rings

In this article we investigate when a complete ideal in a two-dimensional regular local ring is a multiplier ideal of some ideal with an integral multiplying parameter. In particular, we show that this question is closely connected to the Gorenstein property of the blowup along the ideal.

Good quotients of Mori dream spaces

We show that good quotients of algebraic varieties with finitely generated Cox ring have again finitely generated Cox ring.

伴随Chevalley群的Sha-刚性

Sha-rigidity of adjoint Chevalley groups of types $A_1$, $A_2$, $B_2$, $G_2$ over commutative rings

本文证明了在特定环上，A1、A2、B2和G2类型的伴随Chevalley群及其初等子群的局部内自同构是内自同构，从而这些群具有Sha-刚性。

Factorially graded rings of complexity one

We consider finitely generated normal algebras over an algebraically closed field of characteristic zero that come with a complexity one grading by a finitely generated abelian group such that the conditions of a UFD are satisfied for homogeneous elements. Our main results describe these algebras in terms of generators and relations. We apply this to write down explicitly the possible Cox rings of normal complete rational varieties with a complexity one torus action.

正特征下 $\mathbb{G}_{a}$-表示的不变环并非总是有限生成

Invariant Rings of $ \mathbb{G}_{a} $-Representations are not always Finitely Generated in Positive Characteristic

本文针对正特征域上 Weitzenböck 猜想，利用 Mukai 策略构造一个六维 $\mathbb{G}_{a}$-表示，并借助环面簇几何证明其不变环不是有限生成的，从而给出该猜想的反例。

Adjoints of ideals in regular local rings

The adjoint of an ideal I in a regular local ring R is the R-ideal adj(I):=H^0(Y, Iω_Y), where f:Y -> Spec(R) is a proper birational map with Y nonsingular and IO_Y invertible, and ω_f is a canonical relative dualizing sheaf. (Such an f is supposed to exist.) The basic conjecture is that I.adj(I^n)=adj(I^{n+1}) whenever n is >= the analytic spread of I. This is a strong version of the Briancon-Skoda theorem, implying all other known versions. It follows from the (conjectural) vanishing of H^i(Y,ω_Y) for all i>0. When R is essentially of finite type over a char. 0 field, that vanishing has been…

On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated

Our interest is a regularity of a minimal singular metric of a line bundle. One main conclusion of our general result in this paper is the existence of continuous Hermitian metrics with semi-positive curvatures on the so-called Zariski's example of a line bundle defined over the blow-up of $\mathbb{P}^2$ at some twelve points. This is an example of a line bundle which is nef, big, not semi-ample, and whose section ring is not finitely generated. We generalize this result to the higher dimensional case when the stable base locus of a line bundle is a smooth hypersurface with a holomorphic tubul…

Finite generation of Cox rings

In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely generated, the variety admits a quotient presentation by a quasitorus, which resembles the quotient construction of the projective space. We discuss the problem of the finite generation of Cox rings from a geometric perspective and provide examples of both the finitely and non-finitely generated cases.

When is the intersection of two finitely generated subalgebras of a polynomial ring also finitely generated?

We study two variants of the following question: "Given two finitely generated subalgebras R_1, R_2 of C[x_1, \ldots, x_n], is their intersection also finitely generated?" We show that the smallest value of $n$ for which there is a counterexample is 2 in the general case, and 3 in the case that R_1 and R_2 are integrally closed. We also explain the relation of this question to the problem of constructing algebraic compactifications of C^n and to the moment problem on semialgebraic subsets of R^n. The counterexample for the general case is a simple modification of a construction of Neena Gupta,…

相对指标有界的 Siegel–Jacobi 形式环不是有限生成的

Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated

本文利用环面 b-除子与凸几何证明固定次数、权指比固定或有界的 Siegel–Jacobi 形式环非有限生成，并证明 Kramer 猜想、恢复 Tai 的渐近维数公式。

最小秩完全对称簇的等变格罗滕迪克环

Equivariant Grothendieck ring of a complete symmetric variety of minimal rank

本文刻画了最小秩伴随对称空间正则紧化的等变格罗滕迪克环，推广了Brion-Joshua的等变周环结果及作者此前关于伴随半单群的工作。

Local cohomology modules of a smooth Z-algebra have finitely many associated primes

Let $R$ be a commutative Noetherian ring that is a smooth $\mathbb Z$-algebra. For each ideal $I$ of $R$ and integer $k$, we prove that the local cohomology module $H^k_I(R)$ has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.

A pure subalgebra of a finitely generated algebra is finitely generated

We prove the following. Let $R$ be a Noetherian ring, $B$ a finitely generated $R$-algebra, and $A$ a pure $R$-subalgebra of $B$. Then $A$ is finitely generated over $R$.

Canonical semi-rings of finite graphs and tropical curves

For a projective curve $C$ and the canonical divisor $K_C$ on $C$, it is classically known that the canonical ring $R(C) = \oplus_{m=0}^\infty H^0(C, m K_C)$ is finitely generated in degree at most three. In this article, we study whether analogous statements hold for finite graphs and tropical curves. For any finite graph $G$, we show that the canonical semi-ring $R(G)$ is finitely generated but that the degree of generators are not bounded by a universal constant. For any hyperelliptic tropical curve $Γ$ with integer edge-length, we show that the canonical semi-ring $R(Γ)$ is not finitely ge…

Test ideals in rings with finitely generated anti-canonical algebras

Many results are known about test ideals and $F$-singularities for ${\bf Q}$-Gorenstein rings. In this paper we generalize many of these results to the case when the symbolic Rees algebra $O_X \oplus O_X(-K_X) \oplus O_X(-2K_X) \oplus ...$ is finitely generated (or more generally, in the log setting for $-K_X - Δ$). In particular, we show that the $F$-jumping numbers of $τ(X, a^t)$ are discrete and rational. We show that test ideals $τ(X)$ can be described by alterations as in Blickle-Schwede-Tucker (and hence show that splinters are strongly $F$-regular in this setting -- recovering a result …

Minimal free resolution of a finitely generated module over a regular local ring

Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R,\n)$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $\n$-stable filtrations ${\mathbb M} $ of $M $ and to compare the Betti numbers of $M$ with those of the associated graded module $ gr_{\mathbb M}(M). $ This approach has the advantage that the same module $M$ can be detected by using different filtrations on it. It provides interesting upper bounds for the Betti numbers and we study the modules for which the extremal values are attained. Am…

A valuation theorem for Noetherian rings

Let A and B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra that contains A. Denote by A' the integral closure of A in B. We show that A' is determined by finitely many unique discrete valuation rings. Our result generalizes Rees' classical valuation theorem for ideals. We also obtain a variant of Zariski's main theorem.

Exterior powers of the adjoint representation and the Weyl ring of $E_8$

I derive explicitly all polynomial relations in the character ring of $E_8$ of the form $χ_{\wedge^k \mathfrak{e}_8} - \mathfrak{p}_{k} (χ_{1}, \dots, χ_{8})=0$, where $\wedge^k \mathfrak{e}_8$ is an arbitrary exterior power of the adjoint representation and $χ_{i}$ is the $i^{\rm th}$ fundamental character. This has simultaneous implications for the theory of relativistic integrable systems, Seiberg-Witten theory, quantum topology, orbifold Gromov-Witten theory, and the arithmetic of elliptic curves. The solution is obtained by reducing the problem to a (large, but finite) dimensional linear …