Pub Date : 2021-11-01DOI: 10.1017/s030500412100061x
{"title":"PSP volume 171 issue 3 Cover and Front matter","authors":"","doi":"10.1017/s030500412100061x","DOIUrl":"https://doi.org/10.1017/s030500412100061x","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"31 1","pages":"f1 - f2"},"PeriodicalIF":0.8,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84971267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-17DOI: 10.1017/S030500412300021X
Hülya Argüz
Abstract Gross and Siebert developed a program for constructing in arbitrary dimension a mirror family to a log Calabi–Yau pair (X, D), consisting of a smooth projective variety X with a normal-crossing anti-canonical divisor D in X. In this paper, we provide an algorithm to practically compute explicit equations of the mirror family in the case when X is obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary, and D is the strict transform of the toric boundary. The main ingredient is the heart of the canonical wall structure associated to such pairs (X, D), which is constructed purely combinatorially, following our previous work with Mark Gross. In the case when we blow up a single hypersurface we show that our results agree with previous results computed symplectically by Aroux–Abouzaid–Katzarkov. In the situation when the locus of blow-up is formed by more than a single hypersurface, due to infinitely many walls interacting, writing the equations becomes significantly more challenging. We provide the first examples of explicit equations for mirror families in such situations.
{"title":"Equations of mirrors to log Calabi–Yau pairs via the heart of canonical wall structures","authors":"Hülya Argüz","doi":"10.1017/S030500412300021X","DOIUrl":"https://doi.org/10.1017/S030500412300021X","url":null,"abstract":"Abstract Gross and Siebert developed a program for constructing in arbitrary dimension a mirror family to a log Calabi–Yau pair (X, D), consisting of a smooth projective variety X with a normal-crossing anti-canonical divisor D in X. In this paper, we provide an algorithm to practically compute explicit equations of the mirror family in the case when X is obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary, and D is the strict transform of the toric boundary. The main ingredient is the heart of the canonical wall structure associated to such pairs (X, D), which is constructed purely combinatorially, following our previous work with Mark Gross. In the case when we blow up a single hypersurface we show that our results agree with previous results computed symplectically by Aroux–Abouzaid–Katzarkov. In the situation when the locus of blow-up is formed by more than a single hypersurface, due to infinitely many walls interacting, writing the equations becomes significantly more challenging. We provide the first examples of explicit equations for mirror families in such situations.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"71 1","pages":"381 - 421"},"PeriodicalIF":0.8,"publicationDate":"2021-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84278174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-25DOI: 10.1017/s0305004123000439
J. Doyle, Vivian Olsiewski Healey, W. Hindes, Rafe Jones
Given a set $S={x^2+c_1,dots,x^2+c_s}$ defined over a field and an infinite sequence $gamma$ of elements of S, one can associate an arboreal representation to $gamma$ , generalising the case of iterating a single polynomial. We study the probability that a random sequence $gamma$ produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets S defined over $mathbb{Z}[t]$ , and we conjecture a similar positive-probability result for suitable sets over $mathbb{Q}$ . As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all S possessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration.
{"title":"Galois groups and prime divisors in random quadratic sequences","authors":"J. Doyle, Vivian Olsiewski Healey, W. Hindes, Rafe Jones","doi":"10.1017/s0305004123000439","DOIUrl":"https://doi.org/10.1017/s0305004123000439","url":null,"abstract":"\u0000 Given a set \u0000 \u0000 \u0000 \u0000$S={x^2+c_1,dots,x^2+c_s}$\u0000\u0000 \u0000 defined over a field and an infinite sequence \u0000 \u0000 \u0000 \u0000$gamma$\u0000\u0000 \u0000 of elements of S, one can associate an arboreal representation to \u0000 \u0000 \u0000 \u0000$gamma$\u0000\u0000 \u0000 , generalising the case of iterating a single polynomial. We study the probability that a random sequence \u0000 \u0000 \u0000 \u0000$gamma$\u0000\u0000 \u0000 produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets S defined over \u0000 \u0000 \u0000 \u0000$mathbb{Z}[t]$\u0000\u0000 \u0000 , and we conjecture a similar positive-probability result for suitable sets over \u0000 \u0000 \u0000 \u0000$mathbb{Q}$\u0000\u0000 \u0000 . As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all S possessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"71 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73196000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-19DOI: 10.1017/S030500412100058X
R. Laterveer
Abstract Let Y be a smooth complete intersection of three quadrics, and assume the dimension of Y is even. We show that Y has a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, the Chow ring of (powers of) Y displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow–Künneth decomposition for double planes.
摘要:设Y为三个二次曲面的光滑完全交，并设Y的维数为偶。在Shen-Vial的意义上，我们证明Y具有乘法的chow - k n次分解。因此，Y的(幂)周氏环表现出类似k3的行为。作为论证的副产品，我们还建立了双平面的乘法周-克第n次分解。
{"title":"Algebraic cycles and intersections of three quadrics","authors":"R. Laterveer","doi":"10.1017/S030500412100058X","DOIUrl":"https://doi.org/10.1017/S030500412100058X","url":null,"abstract":"Abstract Let Y be a smooth complete intersection of three quadrics, and assume the dimension of Y is even. We show that Y has a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, the Chow ring of (powers of) Y displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow–Künneth decomposition for double planes.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"1 1","pages":"349 - 367"},"PeriodicalIF":0.8,"publicationDate":"2021-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82911007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-18DOI: 10.1017/S030500412200010X
Daksh Aggarwal, Unique Subedi, William Verreault, Asif Zaman, Chenghui Zheng
Abstract We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers.
{"title":"Sums of random multiplicative functions over function fields with few irreducible factors","authors":"Daksh Aggarwal, Unique Subedi, William Verreault, Asif Zaman, Chenghui Zheng","doi":"10.1017/S030500412200010X","DOIUrl":"https://doi.org/10.1017/S030500412200010X","url":null,"abstract":"Abstract We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"111 1","pages":"715 - 726"},"PeriodicalIF":0.8,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79173802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-15DOI: 10.1017/S0305004123000270
D. Ghioca, S. Saleh
Abstract We prove the Zariski dense orbit conjecture in positive characteristic for regular self-maps of split semiabelian varieties.
摘要证明了分裂半abel变体正则自映射的正特征的Zariski密轨道猜想。
{"title":"Zariski dense orbits for regular self-maps of split semiabelian varieties in positive characteristic","authors":"D. Ghioca, S. Saleh","doi":"10.1017/S0305004123000270","DOIUrl":"https://doi.org/10.1017/S0305004123000270","url":null,"abstract":"Abstract We prove the Zariski dense orbit conjecture in positive characteristic for regular self-maps of split semiabelian varieties.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"6 1","pages":"479 - 519"},"PeriodicalIF":0.8,"publicationDate":"2021-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89076072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-12DOI: 10.1017/s0305004121000578
{"title":"PSP volume 171 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s0305004121000578","DOIUrl":"https://doi.org/10.1017/s0305004121000578","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73606596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-12DOI: 10.1017/s0305004121000566
{"title":"PSP volume 171 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s0305004121000566","DOIUrl":"https://doi.org/10.1017/s0305004121000566","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"348 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82906387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-03DOI: 10.1017/S0305004122000317
Natalia S. Dergacheva, A. Klyachko
Abstract How many 2-cells must two finite CW-complexes have to admit a common, but not finite common, covering? Leighton’s theorem says that both complexes must have 2-cells. We construct an almost (?) minimal example with two 2-cells in each complex.
{"title":"Small non-Leighton two-complexes","authors":"Natalia S. Dergacheva, A. Klyachko","doi":"10.1017/S0305004122000317","DOIUrl":"https://doi.org/10.1017/S0305004122000317","url":null,"abstract":"Abstract How many 2-cells must two finite CW-complexes have to admit a common, but not finite common, covering? Leighton’s theorem says that both complexes must have 2-cells. We construct an almost (?) minimal example with two 2-cells in each complex.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"8 ","pages":"385 - 391"},"PeriodicalIF":0.8,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72552053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-26DOI: 10.1017/S0305004123000166
T. Wooley
Abstract When $kgeqslant 4$ and $0leqslant dleqslant (k-2)/4$ , we consider the system of Diophantine equations begin{align*}x_1^j+ldots +x_k^j=y_1^j+ldots +y_k^jquad (1leqslant jleqslant k,, jne k-d).end{align*} We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when $d=o!left(k^{1/4}right)$ .
{"title":"Paucity problems and some relatives of Vinogradov’s mean value theorem","authors":"T. Wooley","doi":"10.1017/S0305004123000166","DOIUrl":"https://doi.org/10.1017/S0305004123000166","url":null,"abstract":"Abstract When \u0000$kgeqslant 4$\u0000 and \u0000$0leqslant dleqslant (k-2)/4$\u0000 , we consider the system of Diophantine equations \u0000begin{align*}x_1^j+ldots +x_k^j=y_1^j+ldots +y_k^jquad (1leqslant jleqslant k,, jne k-d).end{align*}\u0000 We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when \u0000$d=o!left(k^{1/4}right)$\u0000 .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"42 1","pages":"327 - 343"},"PeriodicalIF":0.8,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77273162","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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