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Initial degenerations of Grassmannians. (English) Zbl 1470.14122

The Grassmannian \(\mathrm{Gr}(d,n)\) parametrizing the \(d\)-planes inside \(\Bbbk^n\) can be seen as a subvariety of some projective space via the Plücker embedding. Its intersection \(\mathrm{Gr}_0(d,n)\) with the main strata of the projective space consists of \(d\)-planes with non-zero Plücker coordinates. The tropicalization \(T\mathrm{Gr}_0(d,n)\) of \(\mathrm{Gr}_0(d,n)\) is a fan inside \(\Lambda^d\mathbb{R}^n\) that can be seen from several points of view:
First, it is the image by the valuation of the \(\mathbb{K}\)-points of \(G_0(d,n)\) for \(\mathbb{K}\) some algebraically closed valued field with surjective valuation,
Then, it is the moduli space of linear tropical space that are realizable over valued extensions of \(\Bbbk\),
Finally, it consists of the vectors \(w\in\Lambda^d\mathbb{R}^n\) such that the initial degeneration \(\mathrm{in}_w \mathrm{Gr}_0(d,n)\) is non-empty. The goal of the paper is to study the properties of some of the initial degenerations of the Grassmannian.

The cones of \(T \mathrm{Gr}(d,n)\) are in bijection with some subdivisions \(\Delta_w\) of the hypersimplex \(\Delta(d,n)\) into matroid polytopes. Here, \(w\) is some point that belongs to the relative interior of the considered cone. As the ideals giving the initial degenerations \(\mathrm{in}\mathrm{Gr}_0(d,n)\) can be quite difficult to handle, even with a computer, the paper studies them using a closed embedding inside some varieties constructed from \(\Delta_w\) which are easier to manipulate. The description of this variety is as follows.
To each \(d\)-plane \(L\) inside \(\Bbbk^n\) is associated a matroid obtained via the hyperplane arrangement that the coordinate hyperplanes of \(\Bbbk^n\) realize inside the subspace \(L\). Such a matroid is called realizable. Conversely, to each realizable matroid \(M\), we can consider the thin Schubert cell \(\mathrm{Gr}_M\subset\mathrm{Gr}(d,n)\) consisting of subspaces inducing the matroid \(M\).
Consider the subdivision \(\Delta_w\) of \(\Delta(d,n)\) into matroid polytopes. Let \(Q\) be one of the matroid polytopes and let \(M_Q\) be the corresponding matroid. A face \(Q'\) of \(Q\) corresponds to a matroid \(M_{Q'}\) which is the direct sum of a quotient and a contraction of \(M_Q\). Using this description, the paper gives a way to lift the inclusion \(Q'\subset Q\) to an application on the level of thin Schubert cells corresponding to \(M_Q\) and \(M_{Q'}\): \(\mathrm{Gr}_{M_Q}\rightarrow \mathrm{Gr}_{M_{Q'}}\). Therefore, it is possible to consider the inverse limit \[\varprojlim_{Q\in\Delta_w}\mathrm{Gr}_{M_Q},\] associated to the subdivision obtained from a cone of \(T\mathrm{Gr}_0(d,n)\).
For an element \(w\) in the fan \(T \mathrm{Gr}_0(d,n)\), the paper then constructs a closed immersion \[\mathrm{in}_w\mathrm{Gr}_0(d,n)\rightarrow\varprojlim_{Q\in\Delta_w}\mathrm{Gr}_{M_Q}.\] It then uses it to deduce results such as the smoothness and irreducibleness of the initial degenerations of \(\mathrm{Gr}_0(3,7)\), the fact that \(\mathrm{Gr}_0(3,9)\) has a non-connected initial degeneration.
They also use the setting to prove the \(n=7\) case of the following conjecture of Keel and Tevelev [26]. Let \(X(3,n)\) be the normalization of the quotient of \(\mathrm{Gr}(3,n)\) by a maximal torus \(H\subset \mathrm{PGL}(d)\), and \(X_0(3,n)\) the same quotient with \(\mathrm{Gr}_0(3,n)\) instead. As it is already known that \(X(3,n)\) is not log-canonical for \(n\geqslant 9\), the conjecture states that \(X(3,n)\) is a schön and log canonical compactification for \(X_0(3,n)\) when \(n=6,7\) and \(8\). The conjecture was already proven for \(n=6\) by M. Luxton [The log canonical compactification of the moduli space of six lines in \({{\mathbb{P}}}^2\). Texas: University of Texas at Austin (PhD Thesis) (2008)].

MSC:

14T15 Combinatorial aspects of tropical varieties
14M15 Grassmannians, Schubert varieties, flag manifolds
14E25 Embeddings in algebraic geometry
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