Project description
New study bridges algebra, geometry and topology through syzygies
Koszul algebra has played an important role in algebraic topology, commutative algebra and other mathematical fields. Researchers have recently shown that Koszul modules are novel homological objects that establish striking connections between algebraic geometry and geometric group theory. Based on this finding, the same group will study the interconnections between algebraic geometry and geometric group theory using syzygies. EU funding of the SYZYGY project will enable them to prove Green’s conjecture about equations of algebraic curves and compute the Kodaira dimension of the moduli space of curves of genus between 17 and 21. Focus will also be placed on finding algebro-geometric interpretations for the Alexander invariants of the Torelli group.
Objective
This is a proposal aimed at harvesting interconnections between algebraic geometry and geometric group theory using syzygies. The impetus of the proposal is the recent breakthrough in which, inspired by rational homotopy theory, we introduced Koszul modules as novel homological objects establishing striking connections between algebraic geometry and geometric group theory. Deep statements in geometric group theory have startling counterparts in algebraic geometry and these connection led to a recent proof of Green's Conjecture for generic algebraic curves in arbitrary characteristic, as well as to a dramatically simpler proof in characteristic zero. Based on a dynamic view of mathematics in which ideas from one field trigger major developments in another, I propose to lead a group at HU Berlin dedicated to the following major themes, which are outlined in the proposal: (i) Find a solution to Green's Conjecture on the syzygies of an arbitrary smooth canonical curve of genus g. Find a full solution to the Prym-Green Conjecture on the syzygies of a general paracanonical algebraic curve of genus g. Formulate and prove a non-commutative Green's Conjecture for super algebraic curves. (ii) Compute the Kodaira dimension of the moduli space of curves in the transition case from unirationality to general type, when g is between 17 and 21. Construct the canonical model of the moduli space of curves and find its modular interpretation. (iii) Find algebro-geometric interpretations for Alexander invariants of the Torelli group of the mapping class group and that of the Torelli group of the free group. Understand the link between these invariants, the homotopy type and the cohomological dimension of the moduli space of curves. (iv) Get structural insight in the newly discovered topological version of Green's Conjecture involving the Alexander invariant of the group.
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ERC-ADG - Advanced GrantHost institution
10117 Berlin
Germany