Arc Space Research Articles

Divisorial Valuations via Arcs

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.

Families of higher dimensional germs with bijective Nash map

Let (X,0) be a germ of complex analytic normal variety, non-singular outside 0. An essential divisor over (X,0) is a divisorial valuation of the field of meromorphic functions on (X,0), whose center on any resolution of the germ is an irreducible component of the exceptional locus. The Nash map associates to each irreducible component of the space of arcs through 0 on X the unique essential divisor intersected by the strict transform of the generic arc in the component. Nash proved its injectivity and asked if it was bijective. We prove that this is the case if there exists a divisorial resolution π of (X,0) such that its reduced exceptional divisor carries sufficiently many π-ample divisors (in a sense we define). Then we apply this criterion to construct an infinite number of families of 3-dimensional examples, which are not analytically isomorphic to germs of toric 3-folds (the only class of normal 3-fold germs with bijective Nash map known before).

A method to search for strong galaxy–galaxy lenses in optical imaging surveys

We present a semi-automated method to search for strong galaxy–galaxy lenses in optical imaging surveys. Our search technique constrains the shape of strongly lensed galaxies (or arcs) in a multiparameter space, which includes the third-order (octopole) moments of objects. This method is applied to the Deep Lens Survey (DLS), a deep ground-based weak lensing survey imaging to R∼ 26. The parameter space of arcs in the DLS is simulated using real galaxies extracted from deep Hubble Space Telescope fields in order to more accurately reproduce the properties of arcs. Arcs are detected in the DLS using a pixel thresholding method and candidate arcs are selected within this multiparameter space. Examples of strong galaxy–galaxy lens candidates discovered in the DLS F2 field (4 deg2) are presented.

Differential arcs and regular types in differential fields

We introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety is determined by its arcs at a point. Using differential arcs, we show that if (K,+,×, δ1, . . . , δn) is a differentially closed field of characteristic zero with n commuting derivations and p ∈ S(K) is a regular type over K, then either p is locally modular or there is a definable subgroup G ≤ (K,+) of the additive group having a regular generic type that is nonorthogonal to p.

On the Motivic Measure on the Space of Functions

Motivic measure on the space of functions was introduced by Campillo, Delgado, and Gusein-Zade as an analog of the motivic measure on the space of arcs. In this paper we prove that the measure on the space of functions can be related to the motivic measure on the space of arcs by a factor, which can be defined explicitly in geometric terms. This provides a possibility to rewrite motivic integrals over the space of functions as integrals over the union of all symmetric powers of the space of arcs.

Motivic Serre invariants, ramification, and the analytic Milnor fiber

We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to the germ of a morphism f from a smooth complex algebraic variety X to the affine line. This analytic Milnor fiber is a smooth rigid variety over the field of Laurent series C((t)). Its etale cohomology coincides with the singular cohomology of the classical topological Milnor fiber of f; the monodromy transformation is given by the Galois action. Moreover, the points on the analytic Milnor fiber are closely related to the motivic zeta function of f, and the arc space of X. We show how the motivic zeta function can be recovered as some kind of Weil zeta function of the formal completion of X along the special fiber of f, and we establish a corresponding Grothendieck trace formula, which relates, in particular, the rational points on the analytic Milnor fiber over finite extensions of C((t)), to the Galois action on its etale cohomology. The general observation is that the arithmetic properties of the analytic Milnor fiber reflect the structure of the singularity of the germ f.

On d-dual hyperovals in

Abstract In P G ( d ( d + 3 ) / 2 , 2 ) , there are three known families of dual hyperovals; Huybrechts's dual hyperovals ([C. Huybrechts, Dimensional dual hyperovals in projective spaces and c.AC* geometries, Discrete math. 255 (2002), 503–532]), Veronesean dual hyperovals ([J. Thas and H. van Maldeghem, Characterizations of the finite quadric Veroneseans V n 2 n , Quart. J. Math. Oxford. 55 (2004), 99–113] and [S. Yoshiara, Ambient spaces of dimensional dual arcs, J. Alg. Combin. 19 (2004), 5–23]), and the family constructed by Buratti and Del Fra ([M. Buratti and A. del Fra, Semi-Boolean quadruple systems and dimensional dual hyperovals, Adv. Geom. 3 (2003), 245–253] and [A. del Fra and S. Yoshiara, Dimensional dual hyperovals associated with Steiner systems, Europ. J. Combinatorics. 26 (2005), 173–194]) based on Huybrechts's dual hyper-ovals. In this note, we construct a family of dual hyperovals based on Veronesean dual hyperovals. We also prove that our dual hyperovals are not isomorphic to the Veronesean dual hyperovals and do not satisfy Property (T ). Hence, we have a new family of dual hyperovals in P G ( d ( d + 3 ) / 2 , 2 ) .

The powder's thermal behavior on the surface of the melting pool during PTA powder surfacing

The thermal behavior of ceramic powders on the surface of a melting pool during the plasma transferred-arc powder surfacing are theoretically analyzed. The results reveal that for a ceramic powder particle, its experienced heat power from melting pool is much greater than that from arc space during the powder PTA surfacing. It means the heating by the melting pool has taken a much more important part in the particle's thermal behavior than the heating by plasma arc column for a ceramic particle. However, different from the heating by the plasma arc column, the particle's heated power on the 100 A melting pool surface is greater than that on the 200 A melting pool surface.

Multiplier ideals of hyperplane arrangements

In this note we compute multiplier ideals of hyperplane arrangements. This is done using the interpretation of multiplier ideals in terms of spaces of arcs by Ein, Lazarsfeld, and Mustaţǎ (2004).

A curve selection lemma in spaces of arcs and the image of the Nash map

We prove a finiteness result in the space of arcs . As a consequence, we extend to all dimensions the problem of wedges, proposed by Lejeune-Jalabert (Arcs analytiques et résolution minimale des singularités des surfaces quasi-homogénes, Lecture Notes in Mathematics, vol. 777 (Springer, Berlin, 1980), 303–336), and we obtain that an affirmative answer to this problem is equivalent to the surjectivity of the Nash map. This implies, for instance, that the Nash map is bijective for sandwiched surface singularities.

Integration over spaces of nonparametrized arcs and motivic versions of the monodromy zeta function

Notions of integration of motivic type over the space of arcs factorized by the natural C*-action and over the space of nonparametrized arcs (branches) are developed. As an application, two motivic versions of the zeta function of the classical monodromy transformation of a germ of an analytic function on ℂd are given that correspond to these notions. Another key ingredient in the construction of these motivic versions of the zeta function is the use of the so-called power structure over the Grothendieck ring of varieties introduced by the authors.

SU-FF-T-408: Regular Lesions Stereotactic Treatment: Physical and Dosimetrical Comparison Between Circular and Mini/micro Multileaf Collimators

Purpose: The comparison of stereotactic treatments for small spherical and ellipsoidal intracranial lesions using three different collimators with single isocenter: the standard circular collimators and two dynamic multileaf collimators (DMLC) with 3 mm and 5 mm leaf thickness, micro and mini DMLC respectively. Method and Materials: Spherical and ellipsoidal lesions of different sizes were designed to evaluate treatments with collimators under consideration. For spherical lesions with diameter from 15 mm up to 35 mm a treatment with 9 arcs of 140 degrees was performed. For ellipsoidal lesions treated with circular collimators, different combinations of treatment variables were studied to achieve 80% isodose surface conformation: number of arcs, gantry start/stop angles, couch angles and arc spacing. This treatment was matched up to 9 arcs with the two DMLCs. For the comparison geometrical (penumbra, difference between the target volume and 80% isodose volume and conformal indexes) and radiobiological (EUD, TCP and NTCP) parameters were used. Results: Circular collimators and micro DMLC behave similarly with spherical targets in terms of conformal indexes and TCP/NTCP, but EUD and 20% isodose volume are greater for micro DMLC. In the case of ellipsoidal target TCP/NTCP are comparable, EUD is greater for micro DMLC but conformal indexes are higher for circular collimators. The 20% isodose volume is less with micro DMLC; the difference, however, is entirely due to the wider penumbra along the deformation axis with circular collimators. The same considerations are valid for targets with diameters ⩾ 35 mm, where mini DMLC becomes competitive versus circular collimator. Conclusion: The analysis shows that micro and mini DMLC are more functional for non spherical lesions. Circular collimators however are widely used and yield sharper dose fall off along directions perpendicular to the deformation axis and thus are preferable in the case of neighbourhood of organs at risk.

Arcs and resolution of singularities

For a certain class of varieties X, including the toric surfaces, we derive a formula for the valuation dX on the arc space Open image in new window of a smooth ambient space Y, in terms of an embedded resolution of singularities. A simple transformation rule yields a formula for the geometric Poincare series. Furthermore, we prove that for this class of varieties, the arithmetic and the geometric Poincare series coincide. We also study the geometric valuation for plane curves.

Contact loci in arc spaces

We give a geometric description of the loci in the arc space defined by order of contact with a given subscheme, using the resolution of singularities. This induces an identification of the valuations defined by cylinders in the arc space with divisorial valuations. In particular, we recover the description of invariants coming from the resolution of singularities in terms of arcs and jets.

Optimal Design of Arrayed Waveguide Grating

This paper describes the optimal design of an AWG spectrum to meet various specifications and improve some physical parameters. The objective function is the norm of the difference between design parameters and target values. To obtain the design parameters, the Fourier model is employed and the design variables arc spacing of array waveguide, width of array waveguide, optical path difference, and focal length. The (1+1) Evolution Strategy is employed as the optimization tool. The optimization procedure is applied to a 16-channel AWG and the optimized design variables will considerably improve the system performance.

The arc space of a toric variety

The Nash problem on arc families is affirmatively answered for a toric variety by Ishii and Kollár's paper which also shows the negative answer for general case. The Nash problem is one of questions about the relation between arc families and valuations. In this paper, the relation is described clearly for a toric variety. The arc space of a toric variety admits an action of the group scheme determined by the torus. Each orbit on the arc space corresponds to a lattice point in the cone and therefore corresponds to a toric valuation. The dominant relation among the orbits is described in terms of the lattice points. As a corollary, we obtain the answer to the embedded version of the Nash problem for an invariant ideal on a toric variety.

アルミニウム合金の交流ＴＩＧ溶接に対する電流波形の影響

Aluminum alloy has been used in various industrial sectors because of such a desirable mechanical or metallurgical property as low density, high tolerance to corrosion and high recycling efficiency. On the welding of aluminum alloys, it is required to remove the oxide layer from the surface which prevents from making a proper weld joint. AC method is used in the TIG arc welding of aluminum alloys where the cathode spot breaks down the oxide layer covering the material surface during the period of the electrode positive (EP). The object of this study is to make clear the influence of AC current waveform on AC TIG welding of aluminum alloys. Especially, the influence of frequency and EP time ratio on the bead profile and cleaning width has been discussed in the present paper. As a result, it is made clear that the power consumption in arc space is an important factor governing the bead size, and the power consumption in EP alternation is strongly related to the cleaning action.

The Nash problem on arc families of singularities

Nash [21] proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that appears on every resolution. He asked if the converse also holds: Does every such exceptional divisor correspond to an arc family? We prove that the converse holds for toric singularities but fails in general.

Jet schemes, log discrepancies and inversion of adjunction

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.

Characteristics of Hollow Cathode Arc Plasma as a Heat Source. Investigation into Melting Mechanism Focusing on Electron Density and Temperature.

A feasibility study has been conducted to determine whether Hollow Cathode Arc (HCA) can be used for a welding heat source in space. The HCA method enables an arc discharge to ignite and maintain under low pressure conditions, where it is extremely difficult for the conventional Gas Tungsten Arc (GTA) method to form an arc discharge. In the previous paper, it has been demonstrated that the melting process by HCA under the condition that the gas flow rate is high (1.7ml/s) is significantly dominated by the diffuse anode and is characterized by shallow penetration, and that the electron temperature is about 1200018000K in arc space. In the present paper, the volt-ampere characteristics and the plasma property of HCA under the condition that the gas flow rate is low (0.17ml/s) have been made clear, and the melting phenomenon have been discussed. The results are summarized as follows, (1) The HCA under the condition that the gas flow rate is low is characterized by high voltage and high electron temperature, and the arc column is concentrated under the cathode.(2) The electron temperature is above 60000K near the arc axis.(3) The electrons with high temperature, that is high energy, contribute to the deep/huge penetration under the condition that the gas flowrate is low.