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Register a new userGabbers presentation lemma for finite fields
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We give a proof of Gabbers presentation lemma for finite fields. We use ideas from Poonens proof of Bertinis theorem to prove this lemma in the special case of open subsets of the affine plane. We then reduce the case of general smooth varieties to this special case.
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Following Schmidt and Strunk, we give a proof of Gabber presentation lemma over a noetherian domain with infinite residue fields.
We give non-torsion counterexamples against the integral Tate conjecture for finite fields. We extend the result due to Pirutka and Yagita for prime numbers 2,3,5 to all prime numbers.
These are notes of my lectures at the summer school Higher-dimensional geometry over finite fields in Goettingen, June--July 2007. We present a proof of Tates theorem on homomorphisms of abelian varieties over finite fields (including the $ell=p$ case) that is based on a quaternion trick. In fact, a a slightly stronger version of those theorems with finite coefficients is proven.
In the field of algebraic systems biology, the number of minimal polynomial models constructed using discretized data from an underlying system is related to the number of distinct reduced Grobner bases for the ideal of the data points. While the theory of Grobner bases is extensive, what is missing is a closed form for their number for a given ideal. This work contributes connections between the geometry of data points and the number of Grobner bases associated to small data sets. Furthermore we improve an existing upper bound for the number of Grobner bases specialized for data over a finite field.
We give a presentation of the plane Cremona group over an algebraically closed field with respect to the generators given by the Theorem of Noether and Castelnuovo. This presentation is particularly simple and can be used for explicit calculations.
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