{"title":"Evolution by Levi form in $mathbb{P}^2$","authors":"G. Tomassini","doi":"10.4171/rlm/948","DOIUrl":"https://doi.org/10.4171/rlm/948","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47804666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Francesco Regazzoni, S. Pagani, Alessandro Cosenza, A. Lombardi, A. Quarteroni
{"title":"A physics-informed multi-fidelity approach for the estimation of differential equations parameters in low-data or large-noise regimes","authors":"Francesco Regazzoni, S. Pagani, Alessandro Cosenza, A. Lombardi, A. Quarteroni","doi":"10.4171/rlm/943","DOIUrl":"https://doi.org/10.4171/rlm/943","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41697165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Optimal horizontal joinability on the Engel group","authors":"Alexander Greshnov","doi":"10.4171/rlm/947","DOIUrl":"https://doi.org/10.4171/rlm/947","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47243044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the hessian map of quartic plane curves is a birational morphism onto its image, thus bringing new evidence for a very interesting conjecture of Ciro Ciliberto and Giorgio Ottaviani. Our new approach also yields a simpler proof of the similar property for cubic surfaces, which is already known by the work of these two authors.
{"title":"On the birationality of the Hessian maps of quartic curves and cubic surfaces","authors":"A. Dimca, Gabriel Sticlaru","doi":"10.4171/rlm/991","DOIUrl":"https://doi.org/10.4171/rlm/991","url":null,"abstract":"We show that the hessian map of quartic plane curves is a birational morphism onto its image, thus bringing new evidence for a very interesting conjecture of Ciro Ciliberto and Giorgio Ottaviani. Our new approach also yields a simpler proof of the similar property for cubic surfaces, which is already known by the work of these two authors.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42309343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the setting of the Euclidean space equipped with an arbitrary Radon measure, we prove the equivalence between several notions of function of bounded variation present in the literature. We also study the relation between various definitions of 1-Sobolev function.
{"title":"A note on BV and 1-Sobolev functions on the weighted Euclidean space","authors":"M. Gelli, Danka Luvci'c","doi":"10.4171/rlm/988","DOIUrl":"https://doi.org/10.4171/rlm/988","url":null,"abstract":"In the setting of the Euclidean space equipped with an arbitrary Radon measure, we prove the equivalence between several notions of function of bounded variation present in the literature. We also study the relation between various definitions of 1-Sobolev function.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49374279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is possible to define a continued fraction expansion of elements in a function field of a curve by expanding as a Laurent series in a local parameter. Considering the square root of a polynomial $sqrt{D(t)}$ leads to an interesting theory related to polynomial Pell equations. Unlike the classical Pell equation, the corresponding polynomial equation is not always solvable and its solvability is related to arithmetic conditions on the Jacobian (or generalized Jacobian) of the curve defined by $y^2=D(t)$. In this setting, it has been shown by Zannier in cite{zannier} that the sequence of the degrees of the partial quotients of the continued fraction expansion of $sqrt{D(t)}$ is always periodic, even when the expansion itself is not. In this article we work out in detail the case in which the curve $y^2=D(t)$ has genus 0, establishing explicit geometric conditions corresponding to the appearance of partial quotients of certain degrees in the continued fraction expansion. We also show that there are non-trivial polynomials $D(t)$ with non-periodic expansions such that infinitely many partial quotients have degree greater than one.
{"title":"Hyperelliptic continued fractions in the singular case of genus zero","authors":"Francesco Ballini, F. Veneziano","doi":"10.4171/rlm/989","DOIUrl":"https://doi.org/10.4171/rlm/989","url":null,"abstract":"It is possible to define a continued fraction expansion of elements in a function field of a curve by expanding as a Laurent series in a local parameter. Considering the square root of a polynomial $sqrt{D(t)}$ leads to an interesting theory related to polynomial Pell equations. Unlike the classical Pell equation, the corresponding polynomial equation is not always solvable and its solvability is related to arithmetic conditions on the Jacobian (or generalized Jacobian) of the curve defined by $y^2=D(t)$. In this setting, it has been shown by Zannier in cite{zannier} that the sequence of the degrees of the partial quotients of the continued fraction expansion of $sqrt{D(t)}$ is always periodic, even when the expansion itself is not. In this article we work out in detail the case in which the curve $y^2=D(t)$ has genus 0, establishing explicit geometric conditions corresponding to the appearance of partial quotients of certain degrees in the continued fraction expansion. We also show that there are non-trivial polynomials $D(t)$ with non-periodic expansions such that infinitely many partial quotients have degree greater than one.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43592077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compute the integral cohomology groups of the smooth Brill-Noether varieties G d (C), parametrizing linear series of degree d and dimension exactly r on a general curve C. As an application, we determine the whole intersection cohomology of the singular Brill-Noether loci W r d (C), parametrizing complete linear series on C of degree d and dimension at least r.
我们计算了一般曲线C上光滑Brill-Noether变体G d (C)的整上同调群，并将d次、维数为r的线性序列参数化，作为应用，我们确定了奇异Brill-Noether轨迹W r d (C)的整交上同调，确定了C上d次、维数至少为r的完全线性序列参数化。
{"title":"Betti numbers of Brill–Noether varieties on a general curve","authors":"Camilla Felisetti, C. Fontanari","doi":"10.4171/rlm/964","DOIUrl":"https://doi.org/10.4171/rlm/964","url":null,"abstract":"We compute the integral cohomology groups of the smooth Brill-Noether varieties G d (C), parametrizing linear series of degree d and dimension exactly r on a general curve C. As an application, we determine the whole intersection cohomology of the singular Brill-Noether loci W r d (C), parametrizing complete linear series on C of degree d and dimension at least r.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41708032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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