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We show that delta invariant of a log Fano pair can be approximated by lc places of plt complements if it is no greater than one. Under the assumption that delta invariant (no greater than one) of a log Fano pair can be approximated by lc places of bounded plt complements, we show the existence of divisorial valuation computing delta invariant of this log Fano pair.

Let R be a Cohen–Macaulay local ring, I a strongly Cohen–Macaulay ideal of R. We show that $$R/\text{ Ann }\,_R(I)$$ is a maximal Cohen–Macaulay R-module by means of the delta-invariant. Also it is shown that there exists a Cohen–Macaulay ideal J of R such that the $$\delta _{R/J}$$ -invariant of all Koszul homologies of R with respect I is zero.

We completely determine which simply connected rational homology 5-spheres admit Sasaki–Einstein metrics.

We show that delta invariant is a continuous function on the big cone. We will also introduce an analytic delta invariant in terms of the optimal exponent in the Moser–Trudinger inequality and prove that it varies continuously in the Kahler cone, from which we will deduce the continuity of the greatest Ricci lower bound. Then building on the work Berman–Boucksom–Jonsson, we obtain a uniform Yau–Tian–Donaldson theorem for twisted Kahler–Einstein metrics in transcendental cohomology classes.

Let R be a Cohen-Macaulay local ring with a canonical module. We consider Auslander's (higher) delta invariants of powers of certain ideals of R. Firstly, we shall provide some conditions for an ideal to be a parameter ideal in terms of delta invarints. As an application of this result, we give upper bounds for orders of Ulrich ideals of R when R has Gorenstein punctured spectrum. Secondly, we extend the definition of indices to the ideal case, and generalize the result of Avramov-Buchweitz-Iyengar-Miller on the relationship between the index and regularity., 8 pages

We determine the asymptotic behavior of the Arakelov metric, the Arakelov-Green's function, and the Faltings delta-invariant for arbitrary one-parameter families of complex curves with semistable degeneration. The leading terms in the asymptotics are given a combinatorial interpretation in terms of S. Zhang's theory of admissible Green's functions on polarized metrized graphs., 50 pages

We prove a product formula for $\delta$-invariant and as an application, we show that product of K-(semi, poly)stable Fano varieties is also K-(semi, poly)stable., Comment: 15 pages. Final version

Let C be an algebraic space curve defined by a rational parametrization P(t)∈K(t)ℓ, ℓ≥2. In this paper, we consider the T-function, T(s), which is a polynomial constructed from P(t) by means of a univariate resultant, and we show that T(s) contains essential information concerning the singularities of C. More precisely, we prove that T(s)=∏i=1nHPi(s), where Pi, i=1,…,n, are the (ordinary and non-ordinary) singularities of C and HPi, i=1,…,n, are polynomials, each of them associated to a singularity, whose factors are the fiber functions of those singularities as well as those other belonging to their corresponding neighborhoods. That is, HQ(s)=HQ(s)m−1∏j=1kHQj(s)mj−1, where Q is an m-fold point, Qj,j=1,…,k, are the neighboring singularities of Q, and mj,j=1,…,k, are their corresponding multiplicities (HP denotes the fiber function of P). Thus, by just analyzing the factorization of T, we can obtain all the singularities (ordinary and non-ordinary) as well as interesting data relative to each of them, like its multiplicity, character, fiber or number of associated tangents. Furthermore, in the case of non-ordinary singularities, we can easily get the corresponding number of local branches and delta invariant., Agencia Estatal de Investigación

Given an irreducible surface germ (X, 0) ⊂ (ℂ3, 0) with a one-dimensional singular set Σ, we denote by δ1 (X, 0) the delta invariant of a transverse slice. We show that δ1 (X, 0) ≥ m0 (Σ, 0), with equality if and only if (X, 0) admits a corank 1 parametrization f :(ℂ2, 0) → (ℂ3, 0) whose only singularities outside the origin are transverse double points and semi-cubic cuspidal edges. We then use the local Euler obstruction Eu(X, 0) in order to characterize those surfaces that have finite codimension with respect to -equivalence or as a frontal-type singularity.

In his book on Pseudo-Riemannian geometry, δ -invariants and applications, B.Y. Chen introduced a sequence of curvature invariants. Each of these invariants is used to obtain a lower bound for the length of the mean curvature vector for an immersion in a real space form. A submanifold is called an ideal submanifold, for that curvature invariant, if and only if it realizes equality at every point. The first such introduced invariant is called δ ( 2 ) . On the other hand, a well known notion for submanifolds of Sasakian space forms, is the notion of a contact CR-submanifold. In this paper we combine both notions and start the study of minimal contact CR-submanifolds which are δ ( 2 ) ideal. We relate this to a special class of surfaces and obtain a complete classification in arbitrary dimensions.

The trigonal curves of genus 5 can be represented by projective plane quintics that have 1 singularity of delta invariant 1. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this then gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5., 23 pages, improved clarity

The contents of this dissertation are various in nature and background. A connection between them is a common effort to measure the singularities of a commutative Noetherian ring using homological tools, and associated numerical invariants. First, we study the index of a Gorenstein local ring, a numerical invariant that is defined in terms of Auslander's delta invariant. In particular, we find a counterexample to a conjecture of Songqing Ding relating the index and the minimal Loewy length of an Artinian reduction of the ring. Successively, we focus on two numerical invariants for rings of prime characteristic, the F-pure threshold and the diagonal F-threshold. We relate them with a third number, the a-invariant, proving most of a conjecture made by Hirose, Watanabe, and Yoshida. Finally, we study Golod rings. We present an example of a quotient of a polynomial ring over a field by a product of monomial ideals that is not Golod. This answers, in negative, a question of Welker.

We consider different notions of non-degeneracy, as introduced by Kouchnirenko (NND), Wall (INND) and Beelen-Pellikaan (WNND) for plane curve singularities {f(x,y)=0} and introduce the new notion of weighted homogeneous Newton non-degeneracy (WHNND). It is known that the Milnor number μ resp. the delta-invariant δ can be computed by explicit formulas μN resp. δN from the Newton diagram of f if f is NND resp. WNND. It was however unknown whether the equalities μ=μN resp. δ=δN can be characterized by a certain non-degeneracy condition on f and, if so, by which one. We show that μ=μN resp. δ=δN is equivalent to INND resp. WHNND and give some applications and interesting examples related to the existence of “wild vanishing cycles”. Although the results are new in any characteristic, the main difficulties arise in positive characteristic.

A computation of the constant appearing in the spin-1 bosonization formula is given. This constant relates Faltings’ delta invariant to the zeta-regularized determinant of the Laplace operator with respect to the Arakelov metric.

In this paper we give new explicit formulas for Faltings' $\delta$-invariant in terms of integrals of theta functions, and we deduce an explicit lower bound for $\delta$ only in terms of the genus and an explicit upper bound for the Arakelov-Green function in terms of $\delta$. Furthermore, we give a canonical extension of $\delta$ and the Zhang-Kawazumi invariant $\varphi$ to the moduli space of indecomposable principally polarised complex abelian varieties., Comment: 46 pages. The final publication will appear in Inventiones Mathematicae and is available at link.springer.com/article/10.1007/s00222-016-0713-1. This paper replaces the older paper arXiv:1505.04740, which only treats a special case

This paper deals with the local study of invariants of analytic curve singularities. For monomial curves we obtain a numerical description of the \({\mathcal{A}}_e\) -codimension of parametrized curves in terms of classical invariants of the theory of curves, like the delta invariant, the Tjurina number and the Cohen-Macaulay type of its local ring.

Auslander's delta invariant over Gorenstein commutative local rings is generalized to arbitrary local rings via Vogel's construction of Tate (co)homology and a vanishing theorem is proved. Based on one of the new invariants, Auslander's index of a Gorenstein local ring is generalized to arbitrary local rings and the finiteness of the new index is established.

We explicitly bound the Faltings height of a curve over Q polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the dualizing sheaf. Our results allow us to explicitly bound Arakelov invariants of modular curves, Hurwitz curves and Fermat curves in terms of their genus. Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin algorithm to compute coefficients of modular forms for congruence subgroups of SL2(Z) runs in polynomial time under the Riemann hypothesis for zeta-functions of number fields. This was known before only for certain congruence subgroups. Finally, we use our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a cover of the projective line with fixed branch locus. Our proof uses Merkl's method for bounding Arakelov-Green's functions to deal with archimedean contributions. In the appendix, Peter Bruin proves an explicit version of Merkl's results., 44 pages. Appendix by Peter Bruin. To be published in Algebra and Number Theory

We consider generalizations of Szpiro's classical discriminant conjecture to hyperelliptic curves over a number field $K$, and to smooth, projective and geometrically connected curves $X$ over $K$ of genus at least one. The main results give effective exponential versions of the generalized conjectures for some curves, including all curves $X$ of genus one or two. We obtain in particular exponential versions of Szpiro's classical discriminant conjecture for elliptic curves over $K$. In course of our proofs we establish explicit results for certain Arakelov invariants of hyperelliptic curves (e.g. Faltings' delta invariant) which are of independent interest. The proofs use the theory of logarithmic forms and Arakelov theory for arithmetic surfaces., Comments are very welcome

Let ( R , m ) (R , \mathfrak {m} ) be a Gorenstein complete local ring. Auslander’s higher delta invariants are denoted by δ R n ( M ) \delta _R ^n(M) for each module M M and for each integer n n . We propose a conjecture asking if δ R n ( R / m ℓ ) = 0 \delta _R ^n (R/\mathfrak {m} ^{\ell }) = 0 for any positive integers n n and ℓ \ell . We prove that this is true provided the associated graded ring of R R has depth not less than dim ⁡ R − 1 \operatorname {dim} R -1 . Furthermore we show that there are only finitely many possibilities for a pair of positive integers ( n , ℓ ) (n, \ell ) for which δ R n ( R / m ℓ ) > 0 \delta _R ^n (R/ \mathfrak {m} ^{\ell }) >0 .

We approach the algebraic problem of computing topological invariants for the singularities of a plane complex algebraic curve defined by a squarefree polynomial with inexactly-known coefficients. Consequently, we deal with an ill-posed problem in the sense that, tiny changes in the input data lead to dramatic modifications in the output solution.We present a regularization method for handling the illposedness of the problem. For this purpose, we first design symbolic-numeric algorithms to extract structural information on the plane complex algebraic curve: (i) we compute the link of each singularity by numerical equation solving; (ii) we compute the Alexander polynomial of each link by using algorithms from computational geometry and combinatorial objects from knot theory; (iii) we derive a formula for the delta-invariant and the genus. We then prove the convergence for inexact data of the symbolic-numeric algorithms by using concepts from algebraic geometry and topology.Moreover we perform several numerical experiments, which support the validity for the convergence statement.

The Eisenbud operator of a module over a complete hypersurface ring completely determines the delta invariants of this module.

We study the behavior of the Arakelov-Green's function and Faltings' delta invariant on degenerating Riemann surfaces.

In this paper, necessary and sufficient conditions for the positive invariance of convex polyhedra with respect to linear discrete-time singular systems subject to bounded additive disturbances are established. New notions of delta-invariance under different assumptions on the initial conditions are defined. Specifically, the notions of simple and weak delta-invariance are considered. They can be seen as extensions of the delta-positive invariance concept used for the regular linear systems with additive disturbances. The results are presented by considering classical equivalent system representations for linear singular systems. Apresentam-se condições necessárias e suficientes para a invariância positiva de poliédros convexos relativamente a um sistema singular, linear e em tempo discreto, sujeito a perturbações aditivas e limitadas. Introduz-se as noções de delta-invariância simples e de delta-invariância fraca, associadas a diferentes hipóteses a serem verificadas pelas condições iniciais. Estas noções podem ser consideradas como extensões, para o caso de sistemas singulares, do conceito de delta-invariância utilizado no caso de sistemas regulares sujeitos a perturbações. Os resultados são desenvolvidos a partir de duas formas usuais de representação de sistemas singulares lineares.

There is a complete cohomology theory developed over a commutative noetherian ring in which injectives take the role of projectives in Vogel's construction of complete cohomology theory. We study the interaction between this complete cohomology, that is referred to as [Formula: see text]-complete cohomology, and Vogel's one and give some sufficient conditions for their equivalence. Using [Formula: see text]-complete functors, we assign a new homological invariant to any finitely generated module over an arbitrary commutative noetherian local ring, that would generalize Auslander's delta invariant. We generalize the results about the δ-invariant to arbitrary rings and give a sufficient condition for the vanishing of this new invariant. We also introduce an analogue of the notion of the index of a Gorenstein local ring, introduced by Auslander, for arbitrary local rings and study its behavior under flat extensions of local rings. Finally, we study the connection between the index and Loewy length of a local ring and generalize the main result of [11] to arbitrary rings.