Sequence Of Blow-ups Research Articles

Euler Characteristics of Crepant Resolutions of Weierstrass Models

Based on an identity of Jacobi, we prove a simple formula that computes the pushforward of analytic functions of the exceptional divisor of a blowup of a projective variety along a smooth complete intersection with normal crossing. We apply this pushforward formula to derive generating functions for Euler characteristics of crepant resolutions of singular Weierstrass models given by Tate's algorithm. Since these Euler characteristics depend only on the sequence of blowups and not on the Kodaira fiber itself, nor the associated group, several distinct Tate models have the same Euler characteristic. In the case of elliptic Calabi-Yau threefolds, we also compute the Hodge numbers. For elliptically fibered Calabi-Yau fourfolds, our results also prove a conjecture of Blumenhagen-Grimm-Jurke-Weigand based on F-theory/heterotic string duality.

Cycles of singularities appearing in the resolution problem in positive characteristic

We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity reappears after the last blowup, with just certain exponents of the defining power series shifted upwards. The construction hence yields a cycle. Iterating this cycle leads to an infinite increase of the residual order of the defining power series. This disproves a theorem claimed by Moh about the stability of the residual order under sequences of blowups. It is not a counterexample to the resolution in positive characteristic since larger centers are also permissible and prevent the phenomenon from happening.

Partial resolution by toroidal blow-ups

We answer a question of Keel: How much can one improve a non-toroidal ideal sheaf by a sequence of toroidal blow-ups? Version2: Corollary 5 was proved earlier by Tevelev and Ulirsch.

Profile of blow-up solutions to the exponential Neumann boundary value problem

Arising in problems of prescribed Gaussian and Geodesic curvatures, we consider a blow-up sequence of solutions to the following exponential Neumann boundary value problem: −Δun=Vn(x)|x|2αne2uninBR+,∂un∂ν=hn(x)|x|αneunon∂BR+⋂∂R+2,where ν denotes the outer unit normal vector to ∂BR+⋂∂R+2 and αn→α∈(−1,+∞). We establish a uniform estimate for this blow-up sequence of solutions near an isolated singular blow up point. We treat two cases α∉N and α∈N+ separately, and obtain two different uniform pointwise estimates.

Solitary trajectories of analytic vector fields

A trajectory $$\gamma $$ of an analytic vector fields $$\xi $$ is said to be solitary, if $$\gamma $$ is asymptotic to a formal curve and can be isolated from any other trajectory of $$\xi $$ by a finite sequence of point centered blow ups. We prove that the collection of all solitary trajectories generates a polynomially bounded o-minimal structure.

The super-Toda system and bubbling of spinors

We introduce the super-Toda system on Riemann surfaces and study the blow-up analysis for a sequence of solutions to the super-Toda system on a closed Riemann surface with uniformly bounded energy. In particular, we show the energy identities for the spinor parts of a blow-up sequence of solutions for which there are possibly four types of bubbling solutions, namely, finite energy solutions of the super-Liouville equation or the super-Toda system defined on R2 or on R2∖{0}. This is achieved by showing some new energy gap results for the spinor parts of these four types of bubbling solutions.

Hardy–Sobolev equations with asymptotically vanishing singularity: Blow-up analysis for the minimal energy

We study the asymptotic behavior of a sequence of positive solutions (uϵ)ϵ>0 as ϵ→0 to the family of equations Δuϵ+a(x)uϵ=uϵ2∗(sϵ)−1|x|sϵ in Ωuϵ=0 on ∂Ω.where (sϵ)ϵ>0 is a sequence of positive real numbers such that limϵ→0sϵ=0, 2∗(sϵ)≔2(n−sϵ)n−2 and Ω⊂Rn is a bounded smooth domain such that 0∈∂Ω. When the sequence (uϵ)ϵ>0 is uniformly bounded in L∞, then up to a subsequence it converges strongly to a minimizing solution of the stationary Schrödinger equation with critical growth. In case the sequence blows up, we obtain strong pointwise control on the blow-up sequence, and then using the Pohozaev identity localize the point of singularity, which in this case can at most be one, and derive precise blow-up rates. In particular when n=3 or a≡0 then blow-up can occur only at an interior point of Ω or the point 0∈∂Ω.

Some constraints on positive entropy automorphisms of smooth threefolds

Suppose that $X$ is a smooth, projective threefold over $\mathbb C$ and that $\phi : X \to X$ is an automorphism of positive entropy. We show that one of the following must hold, after replacing $\phi$ by an iterate: i) the canonical class of $X$ is numerically trivial; ii) $\phi$ is imprimitive; iii) $\phi$ is not dynamically minimal. As a consequence, we show that if a smooth threefold $M$ does not admit a primitive automorphism of positive entropy, then no variety constructed by a sequence of smooth blow-ups of $M$ can admit a primitive automorphism of positive entropy. In explaining why the method does not apply to threefolds with terminal singularities, we exhibit a non-uniruled, terminal threefold $X$ with infinitely many $K_X$-negative extremal rays on $NE(X)$.

Refined estimates for simple blow-ups of the scalar curvature equation on $S^n$

In their work on a sharp compactness theorem for the Yamabe problem, Khuri, Marques and Schoen apply a refined blow-up analysis (what we call `second order blow-up argument' in this article) to obtain highly accurate approximate solutions for the Yamabe equation. As for the conformal scalar curvature equation on S^n with n > 3, we examine the second order blow-up argument and obtain refined estimate for a blow-up sequence near a simple blow-up point. The estimate involves local effect from the Taylor expansion of the scalar curvature function, global effect from other blow-up points, and the balance formula as expressed in the Pohozaev identity in an essential way.

An unbounded family of log Calabi–Yau pairs

We give an explicit example of log Calabi–Yau pairs that are log canonical and have a linearly decreasing Euler characteristic. This is constructed in terms of a degree two covering of a sequence of blow ups of three dimensional projective bundles over the Segre–Hirzebruch surfaces \mathbb F_n for every positive integer n big enough.

Nash multiplicities and resolution invariants

The Nash multiplicity sequence was defined by M. Lejeune-Jalabert as a non-increasing sequence of integers attached to a germ of a curve inside a germ of a hypersurface. M. Hickel generalized this notion and described a sequence of blow ups which allows us to compute it and study its behavior. In this paper, we show how this sequence can be used to compute some invariants that appear in algorithmic resolution of singularities.

Newton–Okounkov bodies sprouting on the valuative tree

Given a smooth projective algebraic surface X, a point O in X and a big divisor D on X, we consider the set of all Newton-Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E,p) which is infinitely near to O, in the sense that there is a sequence of blowups mapping the smooth, irreducible rational curve E to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton-Okounkov bodies as (E, p) varies, focusing on the case X = P2.

Mass quantization and minimax solutions for Neri's mean field equation in 2D-turbulence

We study the mean field equation derived by Neri in the context of the statistical mechanics description of 2D-turbulence, under a “stochastic” assumption on the vortex circulations. The corresponding mathematical problem is a nonlocal semilinear elliptic equation with exponential type nonlinearity, containing a probability measure P∈M([−1,1]) which describes the distribution of the vortex circulations. Unlike the more investigated “deterministic” version, we prove that Neri's equation may be viewed as a perturbation of the widely analyzed standard mean field equation, obtained by taking P=δ1. In particular, in the physically relevant case where P is non-negatively supported and P({1})>0, we prove the mass quantization for blow-up sequences. We apply this result to construct minimax type solutions on bounded domains in R2 and on compact 2-manifolds without boundary.

Monomial principalization in the singular setting

We generalize an algorithm by Goward for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field. The normal crossings condition considered by Goward is weakened to the condition that components of the generating divisors meet as complete intersections. This leads to a substantial generalization of the notion of monomial scheme; we call the resulting schemes `\textit{regular crossings} (r.c.) \textit{monomial}.' We prove that r.c.~monomial subschemes in arbitrarily singular varieties can be principalized by a sequence of blow-ups at codimension~2 r.c.~monomial centers.

Construction of blow-up sequences for the prescribed scalar curvature equation on $$S^n$$ S n . III. Aggregated and towering blow-ups

Using annular domains introduced in Part II (Leung, Calc Var PDE 46:1–29, 2013), we present constructive results on blow-up sequences of infinite number of solutions for the (prescribed and fixed) conformal scalar curvature equation on \(S^n\) (\(n\ge 6\)), including aggregated and towering blow-ups. The constructions make use of the Lyapunov–Schmidt reduction method, and count on the hyperbolic structure on the collection of standard bubbles.

Two inequalities for a sequence of blow-ups

Two inequalities for a sequence of blow-ups

Two Inequalities for a Sequence of Blow-Ups

Two Inequalities for a Sequence of Blow-Ups

Singular Rationally Connected Surfaces with Nonzero Pluri-Forms

This paper is concerned with projective rationally connected surfaces $X$ with canonical singularities and having non-zero pluri-forms, i.e. $(\Omega_X^1)^{[\otimes m]}$ has non-zero global sections for some m > 0, where $(\Omega_X^1)^{[\otimes m]}$ is the reflexive hull of $(\Omega_X^1)^{\otimes m}$. We show that any such surface can be obtained from a rational ruled surface by a very explicit sequence of blow-ups and blow-downs. Moreover, we interpret the existence of non-zero pluri-forms in terms of semistable reduction.

Counterexamples to local monomialization in positive characteristic

A local monomialization of an algebraic map between varieties along a valuation is a sequence of blowups of nonsingular subvarieties along the valuation in the domain and target of the map such that the resulting map can be expressed by monomial inclusion in a suitable sense. In this paper we give examples showing that a local monomialization does not always exist over fields of positive characteristic. It was shown in earlier work of the author that a local monomialization always exists over fields of characteristic zero.

Point configurations and translations

The spaces of point configurations on the projective line up to the action of \({{\rm SL}(2,\mathbb{K})}\) and its maximal torus are canonically compactified by the Grothdieck–Knudsen and Losev–Manin moduli spaces \({\overline{M}_{0,n}}\) and \({\overline{L}_n}\) respectively. We examine the configuration space up to the action of the maximal unipotent group \({\mathbb{G}_a \subseteq {\rm SL}(2,,\mathbb{K})}\) and define an analogous compactification. For this we first assign a canonical quotient to the action of a unipotent group on a projective variety. Moreover, we show that similar to \({\overline{M}_{0,n}}\) and \({\overline{L}_n}\) this quotient arises in a sequence of blow-ups from a product of projective spaces.