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Register a new userLogarithmic vector fields for quasihomogeneous curve configurations in P^2
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Let A be a union of smooth plane curves C_i, such that each singular point of A is quasihomogeneous. We prove that if C is a smooth curve such that each singular point of A U C is also quasihomogeneous, then there is an elementary modification of rank two bundles, which relates the O_{P^2} module Der(log A) of vector fields on P^2 tangent to A to the module Der(log A U C). This yields an inductive tool for studying the splitting of the bundles Der(log A) and Der(log A U C), depending on the geometry of the divisor A|_C on C.
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The discriminant, $D$, in the base of a miniversal deformation of an irreducible plane curve singularity, is partitioned according to the genus of the (singular) fibre, or, equivalently, by the sum of the delta invariants of the singular points of the fibre. The members of the partition are known as the {it Severi strata}. The smallest is the $delta$-constant stratum, $D(delta)$, where the genus of the fibre is $0$. It is well known, by work of Givental and Varchenko, to be Lagrangian with respect to the symplectic form $Omega$ obtained by pulling back the intersection form on the cohomology of the fibre via the period mapping. We show that the remaining Severi strata are also co-isotropic with respect to $Omega$, and moreover that the coefficients of the expression of $Omega^{wedge k}$ with respect to a basis of $Omega^{2k}(log D)$ are equations for $D(delta-k+1)$, for $k=1,ldots,delta$. These equations allow us to show that for $E_6$ and $E_8$, $D(delta)$ is Cohen-Macaulay (this was already shown by Givental for $A_{2k}$), and that, as far as we can calculate, for $A_{2k}$ all of the Severi strata are Cohen-Macaulay. Our construction also produces a canonical rank 2 maximal Cohen Macaulay module on the discriminant.
We recall first the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the versality properties of $V$, as studied by du Plessis and Wall. Then we show how the bounds on the global Tjurina number of $V$ obtained by du Plessis and Wall lead to substantial improvements of our previous results on the stability of the reflexive sheaf $Tlangle Vrangle$ of logarithmic vector fields along $V$, and on the Torelli property in the sense of Dolgachev-Kapranov of $V$.
In this paper we characterize the rank two vector bundles on $mathbb{P}^2$ which are invariant under the actions of the parabolic subgroups $G_p:=mathrm{Stab}_p(mathrm{PGL}(3))$ fixing a point in the projective plane, $G_L:=mathrm{Stab}_L(mathrm{PGL}(3))$ fixing a line, and when $pin L$, the Borel subgroup $mathbf{B} = G_p cap G_L$ of $mathrm{PGL}(3)$. Moreover, we prove that the geometrical configuration of the jumping locus induced by the invariance does not, on the other hand, characterize the invariance itself. Indeed, we find infinite families that are almost uniform but not almost homogeneous.
We continue to develop the analytic Langlands program for curves over local fields initiated in arXiv:1908.09677, arXiv:2103.01509 following a suggestion of Langlands and a work of Teschner. Namely, we study the Hecke operators introduced in arXiv:2103.01509 in the case of P^1 over a local field with parabolic structures at finitely many points for the group PGL(2). We establish most of the conjectures of arXiv:1908.09677, arXiv:2103.01509 in this case.
We develop a theory of etale parallel transport for vector bundles with numerically flat reduction on a $p$-adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous $p$-adic representation of the etale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a $p$-adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings $p$-adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.
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