Diagonal degenerations of matrix Schubert varieties

Knutson and Miller (2005) established a connection between the anti-diagonal Gr\"obner degenerations of matrix Schubert varieties and the pre-existing combinatorics of pipe dreams. They used this correspondence to give a geometrically-natural explanation for the appearance of the combinatorially defined Schubert polynomials as representatives of Schubert classes. Recently, Hamaker, Pechenik, and Weigandt proposed a similar connection between diagonal degenerations of matrix Schubert varieties and bumpless pipe dreams, newer combinatorial objects introduced by Lam, Lee, and Shimozono. Hamaker, Pechenik, and Weigandt described new generating sets of the defining ideals of matrix Schubert varieties and conjectured a characterization of permutations for which these generating sets are diagonal Gr\"obner bases. They proved special cases of this conjecture and described diagonal degenerations of matrix Schubert varieties in terms of bumpless pipe dreams in these cases. The purpose of this paper is to prove the general conjecture. The proof uses a connection between liaison and geometric vertex decomposition established in earlier work with Rajchgot.Comment: Comments welcome

Relationships Among Hilbert-Samuel Multiplicities, Koszul Cohomology, and Local Cohomology

We consider relationships among Hilbert-Samuel multiplicities, Koszul cohomology, and local cohomology. In particular, we investigate upper and lower bounds on the ratio e(I,M)/l(M/IM) for m-primary ideals I of the local ring (R,m) and finitely generated quasi-unmixed R-modules M and, in joint work with Linquan Ma, Pham Hung Quy, Ilya Smirnov, and Yongwei Yao, give a precise formula for the upper bound for all finitely-generated R-modules M and show that the ratio is bounded away from 0 whenever M is quasi-unmixed. We also, as independent work, give a characterization of quasi-unmixed R-modules M whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. We show that if M is an equidimensional module over a complete local ring, then M is asymptotically Cohen-Macaulay if and only if the supremum of the set of lengths of lower Koszul cohomology modules on systems of parameters if finite if and only if M is Cohen-Macaulay on the punctured spectrum.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145974/1/triciajk_1.pd

Bumpless pipe dreams encode Gr\"obner geometry of Schubert polynomials

In their study of infinite flag varieties, Lam, Lee, and Shimozono (2021) introduced bumpless pipe dreams in a new combinatorial formula for double Schubert polynomials. These polynomials are the TxT-equivariant cohomology classes of matrix Schubert varieties and of their flat degenerations. We give diagonal term orders with respect to which bumpless pipe dreams index the irreducible components of diagonal Gr\"obner degenerations of matrix Schubert varieties, counted with scheme-theoretic multiplicity. This indexing was conjectured by Hamaker, Pechenik, and Weigandt (2022). We also give a generalization to equidimensional unions of matrix Schubert varieties. This result establishes that bumpless pipe dreams are dual to and as geometrically natural as classical pipe dreams, for which an analogous anti-diagonal theory was developed by Knutson and Miller (2005).Comment: In the original, we made an error in a claimed reduction from arbitrary diagonal term orders. We now only address term orders compatible with droop moves. The case of an arbitrary diagonal term order remains open. We are very grateful to Kuei-Nuan Lin and Yi-Huang Shen for alerting us to a flipped inequality in Lemma 5.10. The proof has been amended. Other more minor improvements throughou

Una Plataforma Web para Gestionar los Derechos de Propiedad Intelectual Resultantes de la Investigación Universitaria

Se presenta el desarrollo de un modelo de procesos y una plataforma web para gestionar los derechos de propiedad industrial asociados a los resultados de los programas y proyectos de investigación que se ejecutan en la Universidad de Talca (Chile). El trabajo se inició con una investigación exploratoria seguida de la construcción de un modelo de procesos haciendo uso de la notación BPMN (Business Process Modeling Notation). Finalmente se procedió a desarrollar la plataforma web aplicando la metodología de desarrollo ágil de proyectos XP (eXtreme Programming). El resultado alcanzado -el modelo de procesos y la plataforma web-, han posibilitado la formalización de procesos, la definición de reglas, la identificación de roles por parte de los distintos actores involucrados, y están facilitando el monitoreo de las solicitudes de protección de los derechos de propiedad industrial en la Universidad

Coordination of protein and mRNA abundances of stromal enzymes and mRNA abundances of the Clp protease subunits during senescence of Phaseolus vulgaris (L.) leaves

Our objective was to determine the coordination of transcript and/or protein abundances of stromal enzymes during leaf senescence. First trifolioliate leaves of Phaseolus vulgaris L. plants were sampled beginning at the time of full leaf expansion; at this same time, half of the plants were switched to a nutrient solution lacking N. Total RNA and soluble protein abundances decreased after full leaf expansion whereas chlorophyll abundance remained constant; N stress enhanced the decline in these traits. Abundances of ribulose-1,5-bisposphate carboxylase/oxygenase (Rubisco; EC 4.1.1.39), Rubisco activase and phosphoribulokinase (Ru5P kinase; EC 2.7.1.19) decreased after full leaf expansion in a coordinated manner for both treatments. In contrast, adenosine diphosphate glucose (ADPGlc) pyrophosphorylase (EC 2.7.7.27) abundance was relatively constant during natural senescence but did decline similar to the other enzymes under N stress. Northern analyses indicated that transcript abundances for all enzymes declined markedly on a fresh-weight basis just after full leaf expansion. This rapid decline was particularly strong for the Rubisco small subunit (rbcS) transcript. The decline was enhanced by N stress for rbcS and Rubisco activase (rca), but not for Ru5P kinase (prk) and ADPGlc pyrophosphorylase (agp). Transcripts of the Clp protease subunits clpC and clpP declined in abundance just after full leaf expansion, similar to the other mRNA species. When Northern blots were analyzed using equal RNA loads, rbcS transcripts still declined markedly just after full leaf expansion whereas rca and clpC transcripts increased over time. The results indicated that senescence was initiated near the time of full leaf expansion, was accelerated by N stress, and was characterized by large decline in transcripts of stromal enzymes. The decreased mRNA abundances were in general associated with steadily declining stromal protein abundances, with ADPGlc pyrophosphorylase being the notable exception. Transcript analyses for the Clp subunits supported a recent report (Shanklin et al., 1995, Plant Cell 7: 1713--1722) indicating that the Clp protease subunits were constitutive throughout development and suggested that ClpC and ClpP do not function as a senescence-specific proteolytic system in Phaseolus