On the construction of \(k\)-regular maps to Grassmannians via algebras of socle dimension two (Q6638228)

Let \(Gr(\tau,N)\) be the complex Grassmann manifold of \(\tau\)-dimensional subspaces in \(\mathbb C^N\). A map \(f:\mathbb C^n\rightarrow Gr(\tau,N)\) is said to be \(k\)-regular if for any \(k\) distinct elements \(x_1,x_2,\ldots,x_k\in\mathbb C^n\) the dimension of the subspace \(f(x_1)+f(x_2)+\cdots+f(x_k)\subseteq\mathbb C^N\) is \(k\cdot\tau\).\N\NThe paper deals with the following problem: for given positive integers \(n\), \(k\) and \(\tau\) find the minimal integer \(N\) such that there exists a \(k\)-regular map \(\mathbb C^n\rightarrow Gr(\tau,N)\). More precisely, the authors provide upper bounds for this integer \(N\) by constructing \(k\)-regular maps using the machinery of algebraic geometry. The main result reads as follows:\N\begin{align*}\NN&\le(n-1)(k-1)+\tau k, \mbox{ if } k\le8 \mbox{ or if } \tau\le2 \mbox{ and } k\le11;\\\NN&\le n(k-1)-1+\tau k, \mbox{ otherwise}.\N\end{align*}\NThe construction goes along the same lines as in [\textit{J. Buczyński} et al., J. Eur. Math. Soc. (JEMS) 21, No. 6, 1775--1808 (2019; Zbl 1418.53005)], where essentially the same problem was studied in the case \(\tau=1\). The rough idea of the construction is the following: the authors take a \(k\)-regular map \(f_0:\mathbb C^n\rightarrow Gr(\tau,N_0)\) for some large \(N_0\) (which is known to exist); they observe a convenient subspace \(W\subset\mathbb C^{N_0}\), which avoids the so called ``bad locus''; and compose \(f_0\) with the map \(Gr(\tau,N_0)\rightarrow Gr(\tau,N_0-\dim W)\) induced by the quotient \(\mathbb C^{N_0}\rightarrow\mathbb C^{N_0}/W\) (the fact that \(W\) does not intersect the ``bad locus'' ensures that the composition is well defined and \(k\)-regular).\N\NIn order to find such a subspace \(W\) with dimension as large as possible, the authors devote the major part of the paper to bounding from above the dimension of the punctual Hilbert scheme \(\mathrm{Hilb}_k(\mathbb A^n,0)\), which is closely related to the ``bad locus''. Along the way they prove that for \(k\le11\) the scheme \(\mathrm{Hilb}_k(\mathbb A^3,0)\) is irreducible of dimension \(2(k-1)\), the result which is easily generalized to an arbitrary threefold \(X\) (and a smooth point \(x\in X\)).