Resumen: We prove that the original compact convex set with no extreme points, constructed by Roberts (1977) is an absolute retract, therefore is homeomorphic to the Hilbert cube. Our proof consists of two parts. In this first part, we give a sufficient condition for a Roberts space to be an AR. In the second part of the paper, we shall apply this
to show that the example of Roberts is an AR.
Palabras clave: Convex set, linear metric space, extreme point, absolute retract
Resumen: In this second part of our paper, we apply the result of Part 1 to show that the compact convex set with no extreme points, constructed by Roberts (1977), is an AR.
Palabras clave: Convex set, linear metric space, extreme point, absolute retract
Resumen: Let Hilb d P 2 =Hilb d P 2 C denote the Hilbert scheme parametrizing 0-dimensional, length-d subschemes of the plane. The authors determine an additive basis for the Chow group A.(Hilb d P 2 ) consisting of classes of closures of locally closed subschemes of P 2 . A key feature of this basis is that it consists of cycle classes whose generic points correspond
to reduced length-d subschemes, making it convenient for use in enumerative problems. This is in contrast to the basis given by G. Ellingsrud and S. A. Strømme that arises from a cell decomposition [Invent Math. 91 (1988), no. 2, 365–370; MR0922805 (89f:14007)]. The authors' method of proof involves showing that their basis specializes (via an intermediate basis) to that of Ellingsrud and Strømme.
As applications of their main results and methods, the authors employ a G m -action on P 2 and the Ellingsrud-Strømme basis for A.(Hilb d P 2 ) to give a basis for the Chow group of the variety W ∗ of Schubert triangles. Their results agree with work of J. Roberts and R. Speiser [Comm. Algebra 15 (1987), no. 9, 1929–1966; MR0898302 (88m:14040)] and A. Collino and W. Fulton [Mém. Soc. Math. France (N.S.) No. 38 (1989), 75–117; MR1044347 (91g:14053)]. The paper concludes with enumerative applications using the authors' basis for A.(Hilb d P 2 ) to verify formulas conjectured by Schubert for the number of bitangent contacts between members of two families of moving plane curves.
Palabras clave: Hilbert scheme; basis for the Chow group
Resumen: Andalusite occurs as an accessory mineral in many types of peraluminous
felsic igneous rocks, including rhyolites, aplites, granites,
pegmatites, and anatectic migmatites. Some published stability
curves for And = Sil and the water-saturated granite solidus permit
a small stability field for andalusite in equilibrium with felsic melts.
We examine 108 samples of andalusite-bearing felsic rocks from
more than 40 localities world-wide. Our purpose is to determine the
origin of andalusite, including the T–P–X controls on andalusite
formation, using eight textural and chemical criteria: size—
compatibility with grain sizes of igneous minerals in the same rock;
shape—ranging from euhedral to anhedral, with no simple correlation
with origin; state of aggregation—single grains or clusters of
grains; association with muscovite—with or without rims of monocrystalline
or polycrystalline muscovite; inclusions—rare mineral
inclusions and melt inclusions; chemical composition—andalusite
with little significant chemical variation, except in iron content
(0.08–1.71 wt % FeO); compositional zoning—concentric, sector,
patchy, oscillatory zoning cryptically reflect growth conditions;
compositions of coexisting phases—biotites with high siderophyllite–
eastonite contents (Alw Ɉ 268 ñ 007 atoms per formula
unit), muscovites with 0.57–4.01 wt % FeO and 0.02–
2.85 wt % TiO2, and apatites with 3.53 ñ 0.18 wt % F.
Coexisting muscovite–biotite pairs have a wide range of F contents,
and FBt = 1.612FMs + 0015. Most coexisting minerals have
compositions consistent with equilibration at magmatic conditions.
The three principal genetic types of andalusite in felsic igneous rocks
are: Type 1 Metamorphic—(a) prograde metamorphic (in thermally
metamorphosed peraluminous granites), (b) retrograde
metamorphic (inversion from sillimanite of unspecified origin) (c) xenocrystic (derivation from local country rocks), and (d) restitic
(derivation from source regions); Type 2 Magmatic—(a) peritectic
(water-undersaturated, T↑) associated with leucosomes in migmatites,
(b) peritectic (water-undersaturated, T↓), as reaction rims on
garnet or cordierite, (c) cotectic (water-undersaturated, T↓) direct
crystallization from a silicate melt, and (d) pegmatitic (watersaturated,
T↓), associated with aplite–pegmatite contacts or pegmatitic
portion alone; Type 3 Metasomatic—(water-saturated,
magma-absent), spatially related to structural discontinuities in
host, replacement of feldspar and/or biotite, intergrowths with
quartz. The great majority of our andalusite samples show one or
more textural or chemical criteria suggesting a magmatic origin. Of
the many possible controls on the formation of andalusite (excess
Al2O3, water concentration and fluid evolution, high Be–B–Li–P,
high F, high Fe–Mn–Ti, and kinetic considerations), the two most
important factors appear to be excess Al2O3 and the effect of
releasing water (either to strip alkalis from the melt or to reduce
alumina solubility in the melt). Of particular importance is the
evidence for magmatic andalusite in granites showing no significant
depression of the solidus, suggesting that the And = Sil equilibrium
must cross the granite solidus rather than lie below it. Magmatic
andalusite, however formed, is susceptible to supra- or sub-solidus
reaction to produce muscovite. In many cases, textural evidence
of this reaction remains, but in other cases muscovite may
completely replace
