Lectures on Invariant Theory

By Igor Dolgachev. 2003

435 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/943a4be3f5fa45e73b5a95de1c0102db40d1979a

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Citations in the corpus

• Compact Moduli Spaces of Del Pezzo Surfaces and K\"ahler-Einstein metrics (2012, background): “Since M4 is naturally isomorphic to the moduli space of binary quintics on P(1), choosing invariants as in [18], Chapter 10.”
• Fixed point loci of moduli spaces of sheaves on toric varieties (2008, unknown): “e have to introduce some minus signs in their formulae for Chern classes and Chern characters before we can apply them. 2Since we want to apply [Dol, Thm. 11.1] and the notion of GIT stable points in [Dol] corresponds to the notion of properly GIT stable points in [MFK] (compare [Dol, Sect. 8.1] and”
• Notes on GIT and symplectic reduction for bundles and varieties (2005, background): “stable ⇐⇒0 ∈G.˜x. x is stable ⇐⇒G.x˜ is closed in Cn+1 and ˜x has finite stabiliser. When G.x˜ is closed, but not necessarily of full dimension, we call x polystable. (This is called “Kempf-stable” in [Dl], “weakly stable” in [Ti4] and plain “stable” in [Do5]. Originally [GIT] the terms for stable an”
• Introduction to actions of algebraic groups (2010, methodology): “We refer to the notes of Woodward for this and further developments; the books by Dolgachev (see [1]) and Mukai (see [8]) may also be recommended, as well as the classic by Mumford, Kirwan and Fogarty (see [9]).”
• Topics in classical algebraic geometry (2006, background): “One can show that the quotient by the duality involution is an open subset of a certain explicitly described 9-dimensional toric variety X (see [91]).”
• Classical Algebraic Geometry: A Modern View (2012, background): “One can show that the quotient by the duality involution is an open subset of a certain explicitly described 9-dimensional toric variety X (see [183]).”
• Higher order Turán inequalities for the partition function (2017, background): “Notice that B,C and B3− 27C2 are actually the invariants of the quartic binary form with the same coefficients as g4,k(x) [16, 20] and 256(B3− 27C2) is the discriminant of g4,k(x).”
• Towards non-reductive geometric invariant theory (2007, background): “aic group G is not reductive then there is an affine G-variety X such that k[X]G is not finitely generated. 3. GIT Basics If X is a normal quasi-projective G-variety, then there exists a G-linearization [11][Theorem 7.3] for the G-action on X; that is, a G-equivariant embedding in TOWARDS NON-REDUCTIVE”
• The topology of moduli spaces of free group representations (2008, background): “Any affine group has a faithful linear representation (see [10] for instance), thus it is a closed subgroup of a general linear group and hence a linear Lie group.”
• Singularities of free group character varieties (2009, unknown): “ Proof. The irreducibles are exactly the GIT stable points (zero dimensional stabilizer and closed orbits) and in general these are Zariski open, which implies the complement is an algebraic set (see [Dol03]). Lemma 3.9. Suppose there exists a set O ⊂ Xr(G)sing∩Xr(G)red that is dense with respect to”