The Number of Generic Singularities

Paper by A. Iarrobin

The family G_T of graded quotients of k[x,y] of given Hilbert function

The nonsingular variety G_T parametrizes all graded ideals I of R=k[x,y] for which the Hilbert function H(R/I)=T. The variety G_T has a natural cellular decomposition: each cell V(E) corresponds to a monomial ideal E for which H(R/E)=T. Given a monomial ideal E, the quotient R/E has a basis of monomials in the shape of a partition P(E), having "diagonal lengths" T. The dimension of the cell V(E) is the number of difference one hooks of P(E). We show that G_T is birational to a certain product SGrass(T) of small Grassman varieties, and that over the complexes, the birational map induces an isomorphism of homology groups, but not usually an isomorphism of rings. We determine the homology ring) when T=(1,2,...,d-1,d,...,d,1). This G(T) is the desingularisation of the d-secant bundle Sec(d,j) of the degree-j rational normal curve, We determine the classes of the pullbacks of the higher singular loci of Sec(d,j). We also use the ring H*(G_T) to determine the number of ideals satisfying an intersection of ramification conditions at different points of P^1. A main tool is a combinatorial "hook code" for the partition P(E), that gives the image of the cell V(E) in SGrass(T).Comment: 37 pages. This version is an extensive rewrite, and includes minor corrections. The main results are the sam

Inverse System of a Symbolic Power II. The Waring Problem for Forms

AbstractThe classical Waring problem for forms is to determine the smallest length s of an additive decomposition of a general degree d homogeneous polynomial or form f in r variables as sum of sdth powers of linear forms. We show that its solution is implied by a result of J. Alexander and A. Hirschowitz, concerning the Hilbert functions of the ideal of functions Vanishing to order two at a generic set of s points in Pr − 1. Using Macaulay′s inverse systems, we show that the Alexander-Hirschowitz result is equivalent to determining the number of linear syzygies of s homogeneous forms in r variables that are dth powers of a given set of general linear forms. We also determine the dimension of the family of degree d forms that have additive decompositions of length s. We then study several notions of length for forms f, having to do with the kind of length-s, zero-dimensional schemes Z in Pr − 1 whose defining ideal I(Z) annihilates the inverse system of f. When Z is to consist of distinct points, we obtain the above length of additive decomposition of f. When Z is smoothable we obtain the "smoothable length" of f; when Z is arbitrary, we obtain a "scheme length" of f. All these lengths are at least as large as the dimension of the vector space of all order-i partial derivates of f, for each i. The above-mentioned length functions are distinct. Using results about the existence of nonsmoothable Gorenstein point singularities in codimension 4, we show that when r = 5 there are forms f of scheme length s, which are not in the closure of the family of forms having additive decompositions of length s. Finally, we propose a new set of Waring problems for forms, using these lengths

Data of unhealthy food availability in hospitals

In our manuscript, we present the food choices available at vending machines in government-run Veterans Affairs Hospitals. The data in this article includes both a quantification of the beverages and packages foods available, along with a comparison of recommendations and sugar content to the government-issued USDA Dietary Guidelines 2015-2020. For further discussion on the results of this study, refer to the full manuscript Lead by Poor Example: An Assessment of Snacks, Soda, and Junk Food Availability in Veterans Affairs Hospitals (Champ et al., 2018) [1]

Apolarity, Hessian and Macaulay polynomials

A result by Macaulay states that an Artinian graded Gorenstein ring R of socle dimension one and socle degree b can be realized as the apolar ring of a homogeneous polynomial f of degree b. If R is the Jacobian ring of a smooth hypersurface g=0, then b is just equal to the degree of the Hessian polynomial of g. In this paper we investigate the relationship between f and the Hessian polynomial of g.Comment: 12 pages. Improved exposition, minor correction

Fermat hypersurfaces and Subcanonical curves

We extend the classical Enriques-Petri Theorem to $s$-subcanonical projectively normal curves, proving that such a curve is $(s+2)$-gonal if and only if it is contained in a surface of minimal degree. Moreover, we show that any Fermat hypersurface of degree $s+2$ is apolar to an $s$-subcanonical $(s+2)$-gonal projectively normal curve, and vice versa.Comment: 18 pages; AMS-LaTe

The Impact of Serum Glucose, Anti-Diabetic Agents, and Statin Usage in Non-Small Cell Lung Cancer Patients Treated With Definitive Chemoradiation

Introduction: Epidemiologic data indicate diabetes confers an augmented risk of lung cancer development, yet the relationship between hyperglycemia, metabolic agents, and prognosis is unclear. We analyzed the impact of hyperglycemia, anti-diabetic agents, and statins on outcomes in non-small cell lung cancer (NSCLC) patients undergoing chemoradiation. Method and Materials: In total, data from 170 patients with stage III NSCLC treated at the University of Pittsburgh Medical Center between 2001 and 2014 were obtained for analysis. Kaplan-Meier survival analysis was used to estimate time-to-event for overall survival (OS), disease-free survival, distant metastasis (DM), and loco-regional control (LRC). Blood glucose values (n = 2870), statins, and diabetic medications were assessed both continuously and categorically in univariable and multivariable Cox proportional hazard regression models to estimate hazard ratios and identify prognostic factors. Results: Tumor volume was a negative prognostic factor for OS, disease-free survival, DM, and LRC (p = 0.001). Tumor stage and treatment time were associated with increased all-cause mortality. Any glucose measurement ≥ 130 mg/dl during treatment (2-year estimate 49.9 vs. 65.8%, p = 0.095) was borderline significant for decreased LRC, with similar trends on multivariable analysis (HR 1.636, p = 0.126) and for OS (HR 1.476, p = 0.130). Statin usage was associated with improved 2-year LRC (53.4 vs. 62.4%, p = 0.088) but not with improvements in survival. Other glycemic parameters, comorbid diabetes diagnosis, or anti-diabetic medications were not significantly associated with outcomes. Conclusions: There were trends for blood glucose value over 130 mg/dl and statin nonuse being associated with inferior prognosis for LRC in stage III NSCLC patients; glycemic state, statin usage, and glucose-modulating medications were not associated with survival outcomes in multivariable analysis in this retrospective database

Initial Results of a Prospective Study of Adjuvant Pancreatic Stereotactic Body Radiation Therapy for Close or Positive Margins

Purpose: Patients with close or positive margins after surgery for pancreatic carcinoma are at a high risk for recurrence. Stereotactic body radiation therapy (SBRT) allows for safe dose escalation with great conformity and short duration of treatment. Herein, we report the initial results of a prospective observational study that evaluated the efficacy and safety of this treatment option. Methods and Materials: Patients eligible for the study had pathologically proven T1-4N0-1M0 pancreatic adenocarcinoma with a positive margin (≤ 1 mm) or a close margin defined as \u3c 2.5 mm. Patients were treated with either neoadjuvant or adjuvant chemotherapy, if eligible for systemic therapy. All patients received 36 Gy in 3 fractions to the close or positive margin site. Results: From February 2013 to January 2018, 50 patients were enrolled with 49 patients treated on protocol and included in the analysis. The median age was 71 years. The median clinical target volume was 11.3 cc and median planning target volume 22.0 cc. The median overall survival was 23.7 months (95% confidence interval, 13.6-33.8). Local progression-free survival at 1 and 2 years was 85% and 77%, respectively. Regional progression-free survival at 1 and 2 years was 73% and 73%, respectively. Distant metastases-free survival was 57% and 49% at 1 and 2 years, respectively. Grade 3+ radiation toxicity was only 4.1% and occurred in 2 patients. Conclusions: Adjuvant pancreatic SBRT was shown to be a safe and feasible treatment option for patients with high-risk pancreatic adenocarcinoma and close or positive margins. This is the first prospective study of SBRT in high-risk postoperative pancreatic cancer. Our results yielded significant local and regional control with low rates of acute toxicity. This technique does not interrupt the administration of systemically dosed multiagent chemotherapy and can be safely interdigitated between cycles because SBRT is only 1 week of treatment

Pure O-sequences and matroid h-vectors

We study Stanley's long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable artinian level monomial ideals, as often done in the past, to the study of properties of pure O-sequences. We propose a conjecture on pure O-sequences and settle it in small socle degrees. This allows us to prove Stanley's conjecture for all matroids of rank 3. At the end of the paper, using our method, we discuss a first possible approach to Stanley's conjecture in full generality. Our technical work on pure O-sequences also uses very recent results of the third author and collaborators.Comment: Contains several changes/updates with respect to the previous version. In particular, a discussion of a possible approach to the general case is included at the end. 13 pages. To appear in the Annals of Combinatoric