اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدTriangular arrangements on the projective plane
146
0
0.0
(
0
)
تأليف
Simone Marchesi
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this work we study line arrangements consisting in lines passing through three non aligned points. We call them triangular arrangements. We prove that any combinatorics of a triangular arrangement is always realized by a Roots-of-Unity-Arrangement, which is a particular class of triangular arrangements. Among these Roots-of Unity-Arrangements we characterize the free ones and show that Teraos conjecture holds for this family. Finally, we give two triangular arrangements having the same weak combinatorics, such that one is free but the other one is not.
قيم البحث
اقرأ أيضاً
We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for projective dime
nsions of hyperplane arrangements. They are generalizations of the free arrangement cases, that can be regarded as the special case of our result when the projective dimension is zero. The keys to prove them are several new methods to determine the surjectivity of the Euler and the Ziegler restriction maps, that is combinatorial when the projective dimension is not maximal for all localizations. Also, we introduce a new class of arrangements in which the projective dimension is comibinatorially determined.
Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $mathcal{A}_I$, the regular nilpotent Hessenberg variety $mbox{Hess}(N,I)$, and the regular semisimple He
ssenberg variety $mbox{Hess}(S,I)$. We show that a certain graded ring derived from the logarithmic derivation module of $mathcal{A}_I$ is isomorphic to $H^*(mbox{Hess}(N,I))$ and $H^*(mbox{Hess}(S,I))^W$, the invariants in $H^*(mbox{Hess}(S,I))$ under an action of the Weyl group $W$ of $G$. This isomorphism is shown for general Lie type, and generalizes Borels celebrated theorem showing that the coinvariant algebra of $W$ is isomorphic to the cohomology ring of the flag variety $G/B$. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map $H^*(G/B)to H^*(mbox{Hess}(N,I))$ announced by Dale Peterson and an affirmative answer to a conjecture of Sommers-Tymoczko are immediate consequences. We also give an explicit ring presentation of $H^*(mbox{Hess}(N,I))$ in types $B$, $C$, and $G$. Such a presentation was already known in type $A$ or when $mbox{Hess}(N,I)$ is the Peterson variety. Moreover, we find the volume polynomial of $mbox{Hess}(N,I)$ and see that the hard Lefschetz property and the Hodge-Riemann relations hold for $mbox{Hess}(N,I)$, despite the fact that it is a singular variety in general.
A general vanishing result for the first cohomology group of affine smooth complex varieties with values in rank one local systems is established. This is applied to the determination of the monodromy action on the first cohomology group of the Milno
r fiber of some line arrangements, including the monomial arrangement and the exceptional reflection arrangement of type $G_{31}$.
We investigate the blow-up of a weighted projective plane at a general point. We provide criteria and algorithms for testing if the result is a Mori dream surface and we compute the Cox ring in several cases. Moreover applications to the study of $overline{M}_{0,n}$ are discussed.
By way of Ziegler restrictions we study the relation between nearly free plane arrangements and combinatorics and we give a Yoshinaga-type criterion for plus-one generated plane arrangements.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
