A vanishing theorem for finitely supported ideals in regular local rings
A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed char- acteristic.) It follows that the adjoint e I of a finitely supported ideal I has order ord e I = sup(ordI +1 −d, 0), and that taking adjoints of finitely supported ideals commutes with taking strict transforms at infinitely near points. In particular, e I is also finitely supported.
Proximity inequalities for complete ideals in two-dimensional regular local rings
Introduction. Among the various algebraic approaches to the classification of singularities of irreducible plane curves (via characteristic pairs, multiplicity sequence, value semigroup, etc.) a particularly attractive one, based on the idea of proximity, was developed by Enriques [4, book 4]. His analysis can be adapted to arbitrary valuations birationally dominating two-dimensional regular local rings, and also to complete—i.e., integrally closed—ideals in such rings. While some motivational material on valuations appears in §5, this paper deals mainly with complete ideals. The principal result, Theorem (2.1), provides a necessary and sufficient condition, the proximity inequalities, for the existence of a complete ideal having a given “point basis.” This is the ideal-theoretic version of an old result on the existence of plane curves with given effective multiplicities at infinitely near points, cf. [4, p. 392, p. 427], [13, p. 196, Thm. 14], [8, p. 49]. Unique factorization for complete ideals and some basic properties of simple complete ideals fall out as corollaries, more or less. Theorem (4.11), on the predecessor of a simple complete ideal, is inspired by [5, §3], but says more. Basically then, our purpose is to publicize the efficacy of the notion of proximity, and in particular to redo some of Zariski’s theory of complete ideals so as to expose further its roots in the classical treatment of the local behavior of linear systems of curves on smooth surfaces .
Variability of Reflectance Measurements with Sensor Altitude and Canopy Type
Data were acquired on canopies of mature corn planted in 76 cm rows, mature soybeans planted in 96 cm rows with 71 percent soil cover, and mature soybeans planed in 76 cm rows with 100 percent soil cover. A LANDSAT band radiometer with a 15 degree field of view was used at ten altitudes ranging from 0.2 m to 10 m above the canopy. At each altitude, measurements were taken at 15 cm intervals also a 2.0 m transect perpendicular to the crop row direction. Reflectance data were plotted as a function of altitude and horizontal position to verify that the variance of measurements at low altitudes was attributable to row effects which disappear at higher altitudes where the sensor integrate across several rows. The coefficient of variation of reflectance decreased exponentially as the sensor was elevated. Systematic sampling (at odd multiples of 0.5 times the row spacing interval) required fewer measurements than simple random sampling over row crop canopies.