scholarly journals Divisors of expected Jacobian type

2021 ◽  
Vol 127 (2) ◽  
pp. 161-184
Author(s):  
Josep Àlvarez Montaner ◽  
Francesc Planas-Vilanova

Divisors whose Jacobian ideal is of linear type have received a lot of attention recently because of its connections with the theory of $D$-modules. In this work we are interested on divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear type and the relation type of its Jacobian ideal coincides with the reduction number with respect to the gradient ideal plus one. We provide conditions in order to be able to describe precisely the equations of the Rees algebra of the Jacobian ideal. We also relate the relation type of the Jacobian ideal to some $D$-module theoretic invariant given by the degree of the Kashiwara operator.

1989 ◽  
Vol 106 (3) ◽  
pp. 445-458 ◽  
Author(s):  
Sam Huckaba

AbstractIf R is a Noetherian local ring and I = (x1, …, xn)R is an ideal of R then the Rees algebra R[It] can be represented as a homomorphic image of the polynomial ring R[Z1, …, Zn]. The kernel is a homogeneous ideal, and the smallest of the degree bounds among all generating sets, called the relation type of I, is independent of the representation. We derive formulae connecting the relation type of I with the reduction number of I when the analytic spread of I exceeds height(I) by one. In the process we define complete d-sequences with respect to I and use them to help achieve our results. In addition some results on the behaviour of the relation type modulo an element are proved, and examples where the relation type is explicitly computed are presented.


2019 ◽  
Vol 19 (09) ◽  
pp. 2050169
Author(s):  
Amir Behzad Farrahy ◽  
Abbas Nasrollah Nejad

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities ([Formula: see text]) is of linear type. We show that any reduced projective quartic curve is of gradient linear type.


2019 ◽  
Vol 124 (2) ◽  
pp. 188-202
Author(s):  
Cleto B. Miranda-Neto

A difficult issue in modern commutative algebra asks for examples of modules (more interestingly, reflexive vector bundles) having prescribed reduction number $r\geq 1$. The problem is even subtler if in addition we are interested in good properties for the Rees algebra. In this note we consider the case $r=1$. Precisely, we show that the module of logarithmic vector fields of the Fermat divisor of any degree in projective $3$-space is a reflexive vector bundle of reduction number $1$ and Gorenstein Rees ring.


1995 ◽  
Vol 38 (3) ◽  
pp. 449-463 ◽  
Author(s):  
M. Herrmann ◽  
C. Huneke ◽  
J. Ribbe

This paper studies questions connected with when the Rees algebra of an ideal or the formring of an ideal is Gorenstein. The main results are for ideals of small analytic deviation, and for m-primary ideals of a regular local ring (R, m). The general point proved is that the Gorenstein property forces (and is sometimes equivalent to) lowering the reduction number of the ideal by one from the value predicted if one only assumes the Rees algebra or formring is Cohen–Macaulay.


Author(s):  
R. Stavetska ◽  
Y. Dynko ◽  
M. Bushtruk ◽  
I. Starostenko ◽  
O. Babenko ◽  
...  

It has been experimentally proven that the strength and direction of the connection between the linear type traits evaluation of the Ukrainian black-and-white dairy breed cows and their body build depends on the type of the build and the sign of the exterior. The study was conducted in 2018 on a breeding plant of the Ukrainian Black-and-White Dairy breed SPC named Shchorsa in Kiev region (n=89). Differentiation of cows on types of their body build was conducted by the method proposed by Chernenko (low-capacity, mid-capacity and high-capacity types) and Kolesnik (by the massiveness index – on the friable and dense, the bone index – delicate and rough, the wide-breast and wide-rump indices – on narrow-body and wide-body types). It was found that the cow-heiferswere well-developed, had intermediate stature and optimum body condition score, deep body, and wide, slight slope rump. They alsohad desirable rear legs, set and foot angle but some toe-out, the udder was well-balanced. Under the good total score of the exterior of studied cows there were some animals with short stature (134– 136 cm), very wide rump, straight rear legs side view and toe-out, extremely steep foot angle, loose fore udder attachment, deep udder below hocks and front teat placement inside of quarter. The degree of development of the studied linear traits of cow-heifers confirms their in-herd variability. The variability of the exterior traits varies from 10,7 % (body depth) to 38,6 % (stature). In addition to the stature of a significant variability was characterized of angularity (22,4 %), rear legs set (22,5 %), body condition score (21,8 %), as well as traits that describe udder: rear udder attachment (31,4 %), udder depth (32,6 %) and front and rear teat position (31,3 % and 25,1 %, respectively). The high variability of linear traits indicates the possibility of effective their selection. As a result of the research, it has been established that the strength and direction of the correlations between the linear type traits evaluation of Ukrainian Black-and-White Dairy breed cows and the type of their body build depends on the type of the body build and the linear trait. The low-, mid- and high-capacity bodies build types (according to Chernenko) reliably correlate with chest width (r=0,58) and body depth (r=0,48), rear legs rear view (r=0,30), the central ligament (r=0,28), udder depth (r =-0,22) and teat length (r=0,25), P˂0,01–0,001. Friable and dense bodies build (according to Kolesnik) were most strongly associated with stature (r=-0,28), chest width (r =-0,70), and body depth (r=-0,38), as well as with rear legs rear view (r=-0,23), P˂0,05...0,001; delicate and rough bodies build – with stature (r=-0,51), angularity (r=-0,22),rump width (r=-0,35), udder depth (r=-0,27) and rear udder attachment (r =0,20), P ˂ 0,05–0,001; narrow-body and wide-body types – with chest width (r=0,30) and rump width (r=0,23), P˂0,05–0,01. Linear type trait evaluation of cows depends on the body build. The strength and direction of the correlations between types of body build and linear traits showed that the most informative is the classification of the body build by low-, mid- and high-capacity according to Chernenko – on the average η2x=17,5 %. The less informative – by narrow-body and wide-body types according to Kolesnik – η2x=7,6 %. The classification of cows on delicate-rough, narrow-body and wide-body types (according to Kolesnik) according to informativeness and reliability can have the intermediate position – η2x=10,0–11,5 %. The strongest influence on the body build has the cows’ height (η2x=27,9 % on average for all studied types of body build), chest width (η2x=33,5 %), body depth (η2x=21,8 %), rump width (η2x=24,1 %) and rear legs (η2x=18,7 %). Key words: cows, linear type trait evaluation, body build, correlation, power of influence.


2020 ◽  
Vol 92 (7) ◽  
pp. 85-89
Author(s):  
L. P. Mendeleeva ◽  
I. G. Rekhtina ◽  
A. M. Kovrigina ◽  
I. E. Kostina ◽  
V. A. Khyshova ◽  
...  

Our case demonstrates severe bone disease in primary AL-amyloidosis without concomitant multiple myeloma. A 30-year-old man had spontaneous vertebral fracture Th8. A computed tomography scan suggested multiple foci of lesions in all the bones. In bone marrow and resected rib werent detected any tumor cells. After 15 years from the beginning of the disease, nephrotic syndrome developed. Based on the kidney biopsy, AL-amyloidosis was confirmed. Amyloid was also detected in the bowel and bone marrow. On the indirect signs (thickening of the interventricular septum 16 mm and increased NT-proBNP 2200 pg/ml), a cardial involvement was confirmed. In the bone marrow (from three sites) was found 2.85% clonal plasma cells with immunophenotype СD138+, СD38dim, СD19-, СD117+, СD81-, СD27-, СD56-. FISH method revealed polysomy 5,9,15 in 3% of the nuclei. Serum free light chain Kappa 575 mg/l (/44.9) was detected. Multiple foci of destruction with increased metabolic activity (SUVmax 3.6) were visualized on PET-CT, and an surgical intervention biopsy was performed from two foci. The number of plasma cells from the destruction foci was 2.5%, and massive amyloid deposition was detected. On CT scan foci of lesions differed from bone lesions at multiple myeloma. Bone fragments of point and linear type (button sequestration) were visualized in most of the destruction foci. The content of the lesion was low density. There was no extraossal spread from large zones of destruction. There was also spontaneous scarring of the some lesions (without therapy). Thus, the diagnosis of multiple myeloma was excluded on the basis based on x-ray signs, of the duration of osteodestructive syndrome (15 years), the absence of plasma infiltration in the bone marrow, including from foci of bone destruction by open biopsy. This observation proves the possibility of damage to the skeleton due to amyloid deposition and justifies the need to include AL-amyloidosis in the spectrum of differential diagnosis of diseases that occur with osteodestructive syndrome.


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