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On the ampleness of the cotangent bundles of complete intersections (1510.06323v2)

Published 21 Oct 2015 in math.AG and math.CV

Abstract: Based on a geometric interpretation of Brotbek's symmetric differential forms, for the intersection family $\mathcal{X}$ of generalized Fermat-type hypersurfaces in $\mathbb{P}{\mathbb{K}}N$ defined over any field $\mathbb K$, we reconstruct explicit symmetric differential forms by applying Cramer's rule, skipping cohomology arguments, and we further exhibit unveiled families of lower degree symmetric differential forms on all possible intersections of $\mathcal{X}$ with coordinate hyperplanes. Thereafter, we develop what we call the `moving coefficients method' to prove a conjecture made by Olivier Debarre: for generic $c\geqslant N/2$ hypersurfaces $H_1,\dots,H_c\subset \mathbb{P}{\mathbb C}N$ of degrees $d_1,\dots,d_c$ sufficiently large, the intersection $X:=H_1 \cap \cdots \cap H_c $ has ample cotangent bundle $\Omega_X$, and concerning effectiveness, the lower bound $ d_1,\dots,d_c\geqslant N{N2} $ works. Lastly, thanks to known results about the Fujita Conjecture, we establish the very-ampleness of $\mathsf{Sym}{\kappa}\,\Omega_X$ for all $\kappa\geqslant 64\, \Big( \sum_{i=1}c\, d_i \Big)2 $.

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