Tangent developable surfaces and the equations defining algebraic curves

Ein, Lawrence; Lazarsfeld, Robert

doi:10.1090/bull/1683

Tangent developable surfaces and the equations defining algebraic curves
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Bull. Amer. Math. Soc. 57 (2020), 23-38 Request permission

Abstract:

This is an introduction, aimed at a general mathematical audience, to recent work of Aprodu, Farkas, Papadima, Raicu, and Weyman. These authors established a long-standing folk conjecture concerning the equations defining the tangent developable surface of the rational normal curve. This in turn led to a new proof of a fundamental theorem of Voisin on the syzygies of generic canonical curves. The present note, which is the write-up of a talk given by the second author at the Current Events seminar at the 2019 JMM, surveys this circle of ideas.

References

Marian Aprodu and Jan Nagel, Koszul cohomology and algebraic geometry, University Lecture Series, vol. 52, American Mathematical Society, Providence, RI, 2010. MR 2573635, DOI 10.1090/ulect/052
Marian Aprodu, Gavril Farkas, Stefan Papadima, Claudiu Raicu and Jerzy Weyman, Kozsul modules and Green’s conjecture, Invent. Math., to appear.
Marian Aprodu, Gavril Farkas, Stefan Papadima, Claudiu Raicu and Jerzy Weyman, Topological invariants of groups and Koszul modules, arxiv:1806.01702
Enrico Arbarello, Maurizio Cornalba, Phillip Griffiths and Joe Harris, The Geometry of Algebraic Curves, Springer Verlag, 1985.
Lawrence Ein and Robert Lazarsfeld, The gonality conjecture on syzygies of algebraic curves of large degree, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 301–313. MR 3415069, DOI 10.1007/s10240-015-0072-2
Lawrence Ein and Robert Lazarsfeld, Syzygies of projective varieties of large degree: recent progress and open problems, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 223–242. MR 3821151, DOI 10.4310/pamq.2005.v1.n2.a8
Lawrence Ein and Robert Lazarsfeld, in preparation.
David Eisenbud, Green’s conjecture: an orientation for algebraists, Free resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990) Res. Notes Math., vol. 2, Jones and Bartlett, Boston, MA, 1992, pp. 51–78. MR 1165318
David Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. A second course in commutative algebra and algebraic geometry. MR 2103875
D. Eisenbud and J. Harris, Divisors on general curves and cuspidal rational curves, Invent. Math. 74 (1983), no. 3, 371–418. MR 724011, DOI 10.1007/BF01394242
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Mark L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125–171. MR 739785
Mark Green and Robert Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), no. 1, 73–90. MR 813583, DOI 10.1007/BF01388754
Robert Lazarsfeld, Brill-Noether-Petri without degenerations, J. Differential Geom. 23 (1986), no. 3, 299–307. MR 852158
Stefan Papadima and Alexander I. Suciu, Vanishing resonance and representations of Lie algebras, J. Reine Angew. Math. 706 (2015), 83–101. MR 3393364, DOI 10.1515/crelle-2013-0073
Frank-Olaf Schreyer, Syzygies of canonical curves and special linear series, Math. Ann. 275 (1986), no. 1, 105–137. MR 849058, DOI 10.1007/BF01458587
Claire Voisin, Courbes tétragonales et cohomologie de Koszul, J. Reine Angew. Math. 387 (1988), 111–121 (French). MR 946352, DOI 10.1515/crll.1988.387.111
Claire Voisin, Green’s generic syzygy conjecture for curves of even genus lying on a $K3$ surface, J. Eur. Math. Soc. (JEMS) 4 (2002), no. 4, 363–404. MR 1941089, DOI 10.1007/s100970200042
Claire Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus, Compos. Math. 141 (2005), no. 5, 1163–1190. MR 2157134, DOI 10.1112/S0010437X05001387