Course Syllabus

LAST UPDATE: 01/28/2024
INSTRUCTOR: Amin Gholampour
WEBPAGE:  https://www.math.umd.edu/~amingh/
OFFICE: MTH 4407 (4th floor in Math building) 
CONTACT: Use ELMS email system or email me at amingh at umd.edu
LECTURES: MWF 12--12:50pm at MTH 0403 
OFFICE HOURS (in-person or virtual):  Make an appointment by email preferably a day before or at least a few hours before the time you would like to meet. Meeting availabilities are in the weekdays and usually during 3-5pm (virtual or in-person), 7-9pm (virtual).
For virtual meeting use the Zoom URL: https://umd.zoom.us/j/5743700841.

ROUPH PLAN:  I plan to discuss topics on algebraic curves and their properties, normalization sequence, genus, algebraic geometry vs. analytic geometry (GAGA principal), nodal curves, dualizing sheaf and duality, Riemann-Roch, stable curves, automorphisms, stable reduction, marked curves, Hilbert schemes, moduli spaces of stable curves as analytic spaces, algebraic spaces, orbifolds and DM stacks, separated- and proper-ness of moduli space, boundary divisors, Picard group of moduli space,  projectivity of moduli space, cycles and tautological classes on moduli spaces, Faber's conjecture, Witten's conjecture, Pixton's conjecture, moduli of stable maps and Gromov-Witten invariants (if time permits).

COURSE  POLICY: There will be sporadic homework assignments and/or in case of interest mini-projects on various related topics that will be gradually introduced as the semester progresses.

LECTURE NOTES: Will be posted weekly in Announcements.
SOME REFERENCES: Geometry of Algebraic Curves I, II by Arbarello et al.
Moduli of Curves by Harris and Morrison.
Lectures on Moduli of Curves by Gieseker.
Geometric Invariant Theory by Mumford et al.
Toroidal Embeddings by Mumford et al.
Algebraic Curves by Fulton.
Algebraic Groups and Class Fields by Serre.
Algebraic Spaces and Stacks by Olsson.
An Invitation to Quantum Cohomology by Kock.
Algebraic Spaces by Artin.
Algebraic spaces by Knutson.
Deformation Theory by Hartshorn.
Lectures on Resolutions of Singularities by Kollar.
The irreducibility of the space of curves of given genus by Deligne and Mumford.
The projectivity of the moduli space of stable covers I, II, III by Knudsen and Mumford.
A conjectural description of the tautological ring of the moduli space of curves by Faber.
Two-dimensional gravity and intersection theory on moduli space by Witten.
Conjectural relations in the tautological ring of Mbar_{g,n} by Pixton.
Notes on stable maps and quantum cohomology by Fulton and Pandharipande.