Toric Manifolds and Geometric Theory of Invariants

Part 1 – Toric Varieties:
Definition and Examples. Fans associated with toric varieties. The cone-orbit correspondence. Resolution of singularities. Divisors and line bundles on toric varieties and their cohomology. Cox rings of toric varieties.
Part 2 – Geometric Invariant Theory (GIT):
Classical invariant theory, Hilbert’s theorem on the finitude of invariants, reductive groups, quotients and orbit spaces, the Hilbert-Mumford stability criterion
Part 3 – GIT Applications:
Construction of moduli spaces, toric varieties and GIT quotients.
Part 4 – Logarithmic geometry:
Basic aspects of the theory.  Application to moduli (if time allows).

References:
William Fulton. – Introduction to Toric Varieties, Princeton University Press, 1993.
David A. Cox, John B. Little, Henry K. Schenck. – Toric Varieties, Graduate Studies in Mathematics, AMS, 2011.
P. E. Newstead. – Introduction to moduli problems and orbit spaces. Tata Lectures on Math. and Physics, vol. 51, Springer, Heidelberg, 1978.
J. Dieudonné and J. Carrell. – Invariant Theory, old and new. Academic Press, New York, 1971. (Also, Adv. Math. 4 (1970), 1–80.
D. Mumford, J. Fogarty, F. Kirwan. – Geometric Invariant Theory, Springer.
D. Gieseker. – Moduli of curves. Tata Lectures on Math. and Physics, vol. 61, Springer, Heidelberg, 1982.