Syzygien und Moduli
Zusammenfassung der Projektergebnisse
The project concerned the study of fundamental questions concerning the structure of the equations and the syzygies of algebraic varieties with applications to moduli problems in algebraic geometry. 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 work 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 most influential conjecture on syzygies has been the one formulated by Mark Green in 1984 describing the structure of the equations of every smooth canonical curve of genus g in terms of the Clifford index of the curve. Progress on Green’s Conjecture guided much of the research on syzygies of algebraic varieties, as well as the development of computer algebra systems like Macaulay. One of the main achievements of the project is the resolution in 2018-2019 of Green’s Conjecture for generic curves in arbitrary characteristic. This work has been published in the journal Inventiones Mathematicae. A second line of work concerns the Kodaira dimension of the moduli space Mg of stable curves of genus g. One of the main results achieved in this period has been the proof that both moduli spaces M22 and M23 are of general type. These are the first moduli spaces of curves to be shown to be of general type in the last 35 years!
Projektbezogene Publikationen (Auswahl)
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The Kodaira dimension of M22 and M23
G. Farkas, D. Jensen & S. Payne
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Koszul modules and Green’s conjecture. Inventiones mathematicae, 218(3), 657-720.
Aprodu, Marian; Farkas, Gavril; Papadima, Ştefan; Raicu, Claudiu & Weyman, Jerzy
