Periodic Reporting for period 3 - SYZYGY (Syzygies, moduli and topological invariants of groups)
Période du rapport: 2023-03-01 au 2024-08-31
The term syzygy originates from astronomy and refers to three celestial bodies lying on a straight line. In mathematics, the term was introduced in 1850 by Sylvester, for whom a syzygy
was a linear relation between certain objects with arbitrary functional coefficients. Though the original applications were in Invariant Theory, it was Hilbert’s landmark paper that ended Invariant Theory in its constructive form and introduced syzygies as objects of pure algebra, creating a new world of free resolutions and higher homological algebra, that was to become hugely influential in
algebraic geometry. Syzygies enable a qualitative understanding of equations, making algebraic geometry accessible to experiment on a scale not seen before. This leads to unexpected patterns and
surprising conjectures that could not have been formulated otherwise.
The impetus of this proposal aims to capitalize on a new perspective emerging from geometric group theory to attack
fundamental problems in algebraic geometry, that include:
1) Finding 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 level l paracanonical algebraic curve of genus g. Determine the entire resolution of a general curve of degree d and genus g embedded in an r-dimensional projective space.
2) Compute the Kodaira dimension of the moduli space Mg of curves of genus g in the intermediate range and therefore determine whether the general curve of genus g in this range can be written down explicitly.
Construct the canonical model of Mg for large g and find its modular interpretation.
3) Find algebro-geometric interpretations for the 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.
4) Use algebraic geometry to determine the Chen ranks of large categories of finitely generated groups.
was a linear relation between certain objects with arbitrary functional coefficients. Though the original applications were in Invariant Theory, it was Hilbert’s landmark paper that ended Invariant Theory in its constructive form and introduced syzygies as objects of pure algebra, creating a new world of free resolutions and higher homological algebra, that was to become hugely influential in
algebraic geometry. Syzygies enable a qualitative understanding of equations, making algebraic geometry accessible to experiment on a scale not seen before. This leads to unexpected patterns and
surprising conjectures that could not have been formulated otherwise.
The impetus of this proposal aims to capitalize on a new perspective emerging from geometric group theory to attack
fundamental problems in algebraic geometry, that include:
1) Finding 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 level l paracanonical algebraic curve of genus g. Determine the entire resolution of a general curve of degree d and genus g embedded in an r-dimensional projective space.
2) Compute the Kodaira dimension of the moduli space Mg of curves of genus g in the intermediate range and therefore determine whether the general curve of genus g in this range can be written down explicitly.
Construct the canonical model of Mg for large g and find its modular interpretation.
3) Find algebro-geometric interpretations for the 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.
4) Use algebraic geometry to determine the Chen ranks of large categories of finitely generated groups.
Farkas, Jensen and Payne using a combination of intersection thery, syzygies and novel tropical methods, have showed that both moduli spaces of curves of genus 22 and 23 are of general type. These are the first moduli space of curves of genus g proven to be of general type in the last 35 years.
In the direction of syzygies of algebraic curves, Farkas and Larson have established the Minimal Resolution Conjecture for points on general curve. Given a suitably general curve of given degree and genus, what are the equation of a general set of points on the curve? There is a lower bound on the number and the complexity of the equations for these points coming in a simple way from the fact that these points lie on this curve and the conjecture asserts that these lower bouds are actually attained. Farkas together with Aprodu, Papadima, Raicu and Weyman have shown how a new perspective on syzygies stemming from topology yields a novel solution to the question of determining the resolution of a general curve of genus g in its embedding given by canonical forms.
In the direction of syzygies of algebraic curves, Farkas and Larson have established the Minimal Resolution Conjecture for points on general curve. Given a suitably general curve of given degree and genus, what are the equation of a general set of points on the curve? There is a lower bound on the number and the complexity of the equations for these points coming in a simple way from the fact that these points lie on this curve and the conjecture asserts that these lower bouds are actually attained. Farkas together with Aprodu, Papadima, Raicu and Weyman have shown how a new perspective on syzygies stemming from topology yields a novel solution to the question of determining the resolution of a general curve of genus g in its embedding given by canonical forms.
It is to be hoped that a Bridgeland stability condition on curves in projective curves can be found that will lead to a description of the canonical model of the moduli space of curves. Similarly, the Kodaira dimension of the moduli space of curves of genus g=17,...,21 should be determined. Concerning the connection with geometric group theory, we will study the resonance varieties and Chen ranks of hyperbolic groups and link the structure of these invariants to algebraic geometry.