Sturmfels Research Articles

On the Equations Defining Some Hilbert Schemes

We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme \(\text {Hilb}^{2} \mathbb {P}^{2}\) of two points on \(\mathbb {P}^{2}\) and the dense open set parametrizing non-planar clusters in the punctual Hilbert scheme \(\text {Hilb}^{4}_{0}(\mathbb {A}^{3})\) of clusters of length four on \(\mathbb {A}^{3}\) with support at the origin. We find explicit equations in projective, respectively affine, embeddings for these spaces. In particular, we answer a question of Bernd Sturmfels who asked for a description of the latter space that is amenable to further computations. While the explicit equations we find are controlled in a precise way by the representation theory of SL3, our arguments also rely on computer algebra.

Towards classifying toric degenerations of cubic surfaces

We investigate the class of degenerations of smooth cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. More precisely, we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and who leave the question of classifying these degenerations in the degree 3 case as an open problem. In order to carry out this classification we describe an approach which is closely related to tropical geometry and present partial results in this direction.

Connected Graphs Arising from Products of Veronese Varieties

Given a Segre squarefree Veronese configuration, following Bernd Sturmfels, we improve the study of the graphs associated to the configuration. We determine two special families of toric ideals and a finite set of moves for each of them, which still guarantee simultaneously the connection of all graphs arising from each family of moves.

Components of Gröbner strata in the Hilbert scheme of points

We fix the lexicographic order $\prec$ on the polynomial ring $S=k[x_{1},...,x_{n}]$ over a ring $k$. We define $\Hi^{\prec\Delta}_{S/k}$, the moduli space of reduced Gr\"obner bases with a given finite standard set $\Delta$, and its open subscheme $\Hi^{\prec\Delta,\et}_{S/k}$, the moduli space of families of $#\Delta$ points whose attached ideal has the standard set $\Delta$. We determine the number of irreducible and connected components of the latter scheme; we show that it is equidimensional over ${\rm Spec}\,k$; and we determine its relative dimension over ${\rm Spec} k$. We show that analogous statements do not hold for the scheme $\Hi^{\prec\Delta}_{S/k}$. Our results prove a version of a conjecture by Bernd Sturmfels.

Algebraic Statistics for Computational Biology. * Edited by Lior Pachter and Bernd Sturmfels

Algebraic Statistics for Computational Biology. * Edited by Lior Pachter and Bernd Sturmfels