The Schur functors and the resolution of determinantal varieties

Abstract

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: Rosa Maria Miró-RoigResolutions is one of the most effective methods to obtain information about varieties in Algebraic Geometry. For many years there has been considerable efforts in finding a resolution of determinantal varieties. To put the problem plainly, assume $R=K[x_{0},...,x_{s}]$ is the polynomial ring over an algebraically closed field of characteristic zero and $\mathbb{P}^{s}$ is the projective space of dimension $s$ over $K$. Given $(r_{i,j})$ a homogeneous matrix of size $pxq$ with entries in $R$, the problem is to find an explicit minimal free resolution of the ideal $I_{t}$ defined by the $txt$ minors of this matrix. Over certain hypothesis on $I_{t}$ , this is a minimal free resolution of the variety $X={z \in\mathbb{P}s|rg((r_{i,j})(z))<t} of \mathbb{P}^s$. It provides the Hilbert polynomial of $X$, the projective dimension and the arithmetically Cohen-Macaulayness of the variety among others characteristics