Ahmad Mokhtar

We study the geometry of the Fano schemes \(\mathrm{F}_k(\mathrm{SD}_n^r)\) of the projective variety \(\mathrm{SD}_n^r\) defined by the \(r\times r\) minors of a symmetric \(n\times n\) matrix filled with indeterminates. These schemes are fine moduli spaces parameterizing \((k+1)\)-dimensional linear spaces of symmetric matrices of rank less than \(r\). We prove that the schemes \(\mathrm{F}_k(\mathrm{SD}_n^r)\) can have generically non-reduced components, and characterize their irreducibility, connectedness, and smoothness. Our approach to connectedness also applies to Fano schemes of rectangular matrices as well as alternating matrices and answers a question of Ilten and Chan. Furthermore, we give a complete description of \(\mathrm{F}_1(\mathrm{SD}_n^r)\) and show that when \(r=n\), the Fano schemes of lines have the expected dimension. We extend some of these results to Fano schemes of alternating matrices of bounded rank. As an application, we provide geometric arguments for several previous results concerning spaces of symmetric and alternating matrices of bounded rank.

We study the geometry of the Fano schemes \(\mathrm{F}_k(\mathrm{SD}_n^r)\) of the projective variety \(\mathrm{SD}_n^r\) defined by the \(r\times r\) minors of a symmetric \(n\times n\) matrix filled with indeterminates. These schemes are fine moduli spaces parameterizing \((k+1)\)-dimensional linear spaces of symmetric matrices of rank less than \(r\). We prove that the schemes \(\mathrm{F}_k(\mathrm{SD}_n^r)\) can have generically non-reduced components, characterize their irreducibility and connectedness, and give results on their smoothness. Our approach to connectedness also applies to Fano schemes of rectangular matrices as well as alternating matrices and answers a question of Ilten and Chan. Furthermore, we give a complete description of \(\mathrm{F}_1(\mathrm{SD}_n^r)\) and show that when \(r=n\), the Fano schemes of lines have the expected dimension. As an application, we provide geometric arguments for several previous results concerning spaces of symmetric matrices of bounded rank.

We study the existence of Khovanskii-finite valuations for rational curves of arithmetic genus two. We provide a semi-explicit description of the locus of degree \(n+2\) rational curves in \(\mathbb{P}^n\) of arithmetic genus two that admit a Khovanskii-finite valuation. Furthermore, we describe an effective method for determining if a rational curve of arithmetic genus two defined over a number field admits a Khovanskii-finite valuation. This provides a criterion for deciding if such curves admit a toric degeneration. Finally, we show that rational curves with a single unibranched singularity are always Khovanskii-finite if their arithmetic genus is sufficiently small.

For any graph \(X\) with the adjacency matrix \(A\), the transition matrix of the continuous-time quantum walk at time \(t\) is given by the matrix-valued function \(\mathcal{H}_X(t)=\mathrm{e}^{itA}\). We say that there is perfect state transfer in \(X\) from the vertex \(u\) to the vertex \(v\) at time \(τ\) if \(|\mathcal{H}_X(τ)_{u,v}| = 1\). It is an important problem to determine whether perfect state transfers can happen on a given family of graphs. In this paper we characterize all the graphs in the Johnson scheme which have this property. Indeed, we show that the Kneser graph \(K(2k,k)\) is the only class in the scheme which admits perfect state transfers. We also show that, under some conditions, some of the unions of the graphs in the Johnson scheme admit perfect state transfer.