The topics in the following syllabus are tentative, and might change. I am still deciding how long each topic will need (most likely more than I have currently given below!).
Check back here often for new versions!
Version 0.6| Lec | Date | Topic | Read | Problem sets | |
| 1 | 2 Sep 2020 | Introduction notes | |||
| 2 | 4 Sep 2020 | Groebner bases, initial ideals and Hilbert functions notes | |||
| 3 | 7 Sep 2020 | Borel fixed ideals and generic initial ideals, I notes | |||
| 4 | 9 Sep 2020 | Borel fixed ideals and generic initial ideals, II notes | |||
| 5 | 11 Sep 2020 | Castelnuovo-Mumford regularity, part 1 notes | graded modules, sheaves, cohomology: users guide |
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| 6 | 14 Sep 2020 | Castelnuovo-Mumford regularity, part 2 notes | defs and basic facts | ||
| 7 | 16 Sep 2020 | Castelnuovo-Mumford regularity, part 3 notes | |||
| 8 | 18 Sep 2020 | Macaulay and Gotzmann theorems, Lex ideals notes | |||
| 9 | 21 Sep 2020 | Macaulay and Gotzmann theorems, Lex ideals notes | |||
| 10 | 23 Sep 2020 | Flatness notes | |||
| 11 | 25 Sep 2020 | Flatness, Groebner degeneration notes | |||
| 12 | 28 Sep 2020 | Flatness and Resolutions Construction of Hilbert scheme (v1) notes |
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| 13 | 30 Sep 2020 | Construction of Hilbert schemes notes | geometry of schemes | Functor of points | |
| 13' | 30 Sep 2020 | Hilbert scheme lunch: Macaulay2 | 11:30 - 12:30 pm | ||
| 14 | 2 Oct 2020 | Construction of Hilbert schemes notes | haiman-sturmfels | Definitions theorems |
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| 15 | 5 Oct 2020 | Haiman-Sturmfels overview notes | maclagan 1999 | Sketch of proof | |
| 16 | 7 Oct 2020 | First order deformations notes | |||
| 17 | 9 Oct 2020 | Zariski tangent spaces of Hilbert schemes | |||
| 18 | 12 Oct 2020 | The lexicographic component | reeves-stillman | ||
| 14 Oct 2020 | no class! | Fall break | |||
| 19 | 16 Oct 2020 | Connectedness, radius of Hilbert scheme |
hartshorne connectedness reeves radius |
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| 20 | 19 Oct 2020 | Local equations and deformation theory 1 | |||
| 21 | 21 Oct 2020 | Local equations and deformation theory 2 | |||
| 22 | 23 Oct 2020 | Piene-Schlessinger part 1 | piene-schlessinger | ||
| 23 | 26 Oct 2020 | Piene-Schlessinger part 2 | |||
| 24 | 28 Oct 2020 | Mumford's example: nonreduced components of Hilb |
mumford 1962 curtin 1981 |
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| 25 | 30 Oct 2020 | Hilbert scheme of points part 1 |
cartwright et al 2009 jelisiejew 2017 jelisiejew 2018 |
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| 26 | 2 Nov 2020 | Hilbert scheme of points part 2 | |||
| 27 | 4 Nov 2020 | Fogarty's theorem: smoothness in dimension 2 | fogarty 1968 | ||
| 28 | 6 Nov 2020 | Skeljnes-Smith: Smooth Hilbert schemes |
skeljnes-smith 2020 staal 2017 ramkumar 2019 |
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| 29 | 9 Nov 2020 | Hilbert scheme of curves in P^3 | |||
| 30 | 11 Nov 2020 | Extremal curves in P^3, Hartshorne-Lella-Schlessinger | hartshorne et al 2012 | ||
| 31 | 13 Nov 2020 | ||||
| 16 Nov 2020 | no class! | semi-finals | |||
| 18 Nov 2020 | no class! | semi-finals | |||
| 20 Nov 2020 | no class! | semi-finals | |||
| 23 Nov 2020 | no class! | semi-finals | |||
| 25 Nov 2020 | no class! | ||||
| 27 Nov 2020 | no class! | Thanksgiving | |||
| 32 | 30 Nov 2020 | ||||
| 33 | 2 Dec 2020 | ||||
| 34 | 4 Dec 2020 | ||||
| 35 | 7 Dec 2020 | ||||
| 36 | 9 Dec 2020 | ||||
| 37 | 11 Dec 2020 | ||||
| 38 | 14 Dec 2020 | Murphy's Law | vakil 2006 | ||
| 39 | 16 Dec 2020 |
