Special vs Weakly-Special Manifolds
Friday, 6.12.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
A fundamental problem in Diophantine Geometry is to characterize geometrically potential density of rational points on an algebraic variety X defined over a number field k, i.e. when the set X(L) is Zariski dense for a finite extension L of k. Abramovich and Colliot-Thélène conjectured that potential density is equivalent to the condition that X is weakly-special, i.e. it does not admit any étale cover that dominates a positive dimensional variety of general type. More recently Campana proposed a competing conjecture using the stronger notion of specialness that he introduced. We will review both conjectures and present results that support Campana’s Conjecture (and program) in the analytic and function field setting. This is joint work with Erwan Rousseau and Julie Wang.\n\n\n
Deformations of Hilbert schemes of points via derived categories
Friday, 13.12.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Hilbert schemes of points on surfaces, and Hilbert squares of higher-dimensional varieties, are important and basic constructions of moduli spaces of sheaves. As such they provide a class of interesting yet tractable varieties. In a joint work with Lie Fu and Theo Raedschelders, we explain how one can (re)prove results about their deformation theory by studying their derived categories, via fully faithful functors and Hochschild cohomology, which describes both classical and noncommutative deformations.
Smoothing Normal Crossing Spaces
Friday, 20.12.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Given a normal crossing variety \(X\), a necessary condition for it to\noccur as the central fiber \(f^{-1}(0)\) of a semistable degeneration \(f:\n\bmathcal{X} \bto \bDelta\) is \(\bmathcal{T}^1_X \bcong \bmathcal{O}_D\) for the\ndouble locus \(D \bsubset X\). Sufficient conditions have been given\nfamously by Friedman for surfaces and by Kawamata-Namikawa in any\ndimension. We give sufficient conditions for smoothing more general\nnormal crossing varieties with \(\bmathcal{T}^1_X\) only globally generated\nby relaxing the condition that the total space \(\bmathcal{X}\) should be\nsmooth. Our main technical tool is the degeneration of a spectral\nsequence in logarithmic geometry that also settles a conjecture of\nDanilov on the cohomology of toroidal pairs.