A uniform Linnik basic lemma and entropy bounds

Del Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory Zachary Alexander Maddock This thesis comprises two parts covering distinct topics in algebraic geometry. In Part I, we construct the first examples of regular del Pezzo surfaces for which the first cohomology group of the structure sheaf is nonzero. Such surfaces, which only exist over imperfect fields, arise as generic fibres of fibrations of singular del Pezzo surfaces in positive characteristic whose total spaces are smooth, and their study is motivated by the minimal model program. We also find a restriction on the integer pairs that are possible as the irregularity (that is, the dimension of the first cohomology group of the structure sheaf) and anti-canonical degree of regular del Pezzo surfaces with positive irregularity. In Part II, we consider a connected reductive group acting linearly on a projective variety over an arbitrary field. We prove a formula that compares intersection numbers on the geometric invariant theory quotient of the variety by the reductive group with intersection numbers on the geometric invariant theory quotient of the variety by a maximal torus, in the case where all semi-stable points are properly stable. These latter intersection numbers involve the top equivariant Chern class of the maximal torus representation given by the quotient of the adjoint representation on the Lie algebra of the reductive group by that of the maximal torus. We provide a purely algebraic proof of the formula when the root system decomposes into irreducible root systems of type A. We are able to remove this restriction on root systems by applying a related result of Shaun Martin from symplectic geometry.

Semisimple Hopf algebras via geometric invariant theory

Theory of computing has given rise to some fundamental mathematical problems, notably the P ≠ NP conjecture, and the related lower bound problems concerning formula or circuit size. We develop an approach to these problems through geometric invariant theory. The goal of this approach is to reduce the hard nonexistence problems under consideration to tractable existence problems.Accordingly, we reduce the arithmetic (characteristic 0) version of the P ≠ NP conjecture, and other related lower bound problems to proving existence of obstructions. These are representations in the homogeneous coordinate rings of orbit-closures in geometric invariant theory [MFK], of a class of points which are partially stable and whose stabilizers have special representation-theoretic properties. However, the Luna-Vust complexity [LV] of these orbit closures is quite high, in contrast with the well-understood homogeneous or almost-homogeneous-spaces, such as G/P [LLM], toric varieties [F3], and spherical embed-dings [BLV], whose Luna-Vust complexity is zero.We take a step towards explicit construction of obstructions by proving two results regarding these orbit closures. The first is a generalization of the Borel-Weil theorem for G/P to these orbit-closures. Second, we conjecture a nice representation-theoretic set of generators for their ideals, and prove a weaker version of the conjecture. Such a set of generators had earlier been given for the ideal of G/P by Lakshmibai, Seshadri, Littelmann [LS, Li3, LLM] and Kostant (cf. [PK]).Finally, using these results, we reduce, in essence, the arithmetic non-existence problems under consideration to fundmental existence and construction problems in representation theory and algebraic geometry that are conjectured to be in the complexity class P. 2000 Mathematics Subject Classification68Q1514L2414L3520G0514M17Key words and phrasesComputational ComplexityGeometric Invariant TheoryRepresentation TheoryClassical GroupsPlethysmHomogeneous Spaces

Quivers, geometric invariant theory, and moduli of linear dynamical systems

We explain how structures analogous to those appearing in the theory of stability conditions on abelian and triangulated categories arise in geometric invariant theory. This leads to an axiomatic notion of a central charge on a scheme with a group action and ultimately to a notion of a stability condition on a stack analogous to that on an abelian category. In the appendix by Ibáñez Núñez, it is explained how central charges can be viewed through the graded points of a stack. We use these ideas to introduce an axiomatic notion of a stability condition for polarized schemes, defined in such a way that K-stability is a special case. In the setting of axiomatic geometric invariant theory on a smooth projective variety, we produce an analytic counterpart to stability and explain the role of the Kempf–Ness theorem. This clarifies many of the structures involved in the study of deformed Hermitian Yang–Mills connections, Z-critical connections and Z-critical Kähler metrics.

Given an infinite reductive algebraic group [Formula: see text], we consider [Formula: see text]-equivariant coherent sheaves with prescribed multiplicities, called [Formula: see text]-constellations, for which two stability notions arise. The first one is analogous to the [Formula: see text]-stability defined for quiver representations by King [Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser.[Formula: see text]2) 45(180) (1994) 515–530] and for [Formula: see text]-constellations by Craw and Ishii [Flops of [Formula: see text]-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124(2) (2004) 259–307], but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for [Formula: see text]-constellations, and depends on some finite subset [Formula: see text] of the isomorphy classes of irreducible representations of [Formula: see text]. We show that these two stability notions do not coincide, answering negatively a question raised in [Becker and Terpereau, Moduli spaces of [Formula: see text]-constellations, Transform. Groups 20(2) (2015) 335–366]. Also, we construct Harder–Narasimhan filtrations for [Formula: see text]-constellations with respect to both stability notions (namely, the [Formula: see text]-HN and [Formula: see text]-HN filtrations). Even though these filtrations do not coincide in general, we prove that they are strongly related: the [Formula: see text]-HN filtration is a subfiltration of the [Formula: see text]-HN filtration, and the polygons of the [Formula: see text]-HN filtrations converge to the polygon of the [Formula: see text]-HN filtration when [Formula: see text] grows.

We study the deformations of the minimally elliptic surface singularity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N 16"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>16</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">N_{16}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A standard argument reduces the study of the deformations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N 16"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>16</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">N_{16}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the study of the moduli space of pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper C comma upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(C,L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of a plane quintic curve and a line. We construct this moduli space in two ways: via the periods of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Baseline 3"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surfaces and by using geometric invariant theory (GIT). The GIT construction depends on the choice of the linearization. In particular, for one choice of linearization we recover the space constructed via <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Baseline 3"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surfaces and for another we obtain the full deformation space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N 16"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>16</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">N_{16}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The two spaces are related by a series of explicit flips. In conclusion, by using the flexibility given by GIT and the standard tools of Hodge theory, we obtain a good understanding of the deformations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N 16"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>16</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">N_{16}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

We study geometric invariant theory (GIT) quotients parameterizing n-pointed conics that generalize the GIT quotients ⁠. Our main result is that admits a morphism to each such GIT quotient, generalizing the well-known result of Kapranov for the simpler quotients. Moreover, these morphisms factor through Hassett’s moduli spaces of weighted pointed rational curves, where the weight data comes from the GIT linearization data.

Torsors, also known as principal bundles or principal homogeneous spaces, are ubiquitous in mathematics. The purpose of this book is to present expository lecture notes and cutting-edge research papers on the theory and applications of torsors and étale homotopy, all written from different perspectives by leading experts. Part one of the book contains lecture notes on recent uses of torsors in geometric invariant theory and representation theory, plus an introduction to the étale homotopy theory of Artin and Mazur. Part two of the book features a milestone paper on the étale homotopy approach to the arithmetic of rational points. Furthermore, the reader will find a collection of research articles on algebraic groups and homogeneous spaces, rational and K3 surfaces, geometric invariant theory, rational points, descent and the Brauer–Manin obstruction. Together, these give a state-of-the-art view of a broad area at the crossroads of number theory and algebraic geometry.

We construct entanglement monotones for multi-qubit states based on Plücker coordinate equations of Grassmann variety, which are a central notion in geometric invariant theory. As an illustrative example, we in detail investigate entanglement monotones of a three-qubit state.

We suggest an approach based on geometric invariant theory to the fundamental lower bound problems in complexity theory concerning formula and circuit size. Specifically, we introduce the notion of a partially stable point in a reductive-group representation, which generalizes the notion of stability in geometric invariant theory due to Mumford [Geometric Invariant Theory, Springer-Verlag, Berlin, 1965]. Then we reduce fundamental lower bound problems in complexity theory to problems concerning infinitesimal neighborhoods of the orbits of partially stable points. We also suggest an approach to tackle the latter class of problems via construction of explicit obstructions.

By work of Looijenga and others, one understands the relationship between Geometric Invariant Theory (GIT) and Baily–Borel compactifications for the moduli spaces of degree-$2$ $K3$surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree-$4$ $K3$surfaces and double Eisenbud–Popescu–Walter (EPW) sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga in order to handle these cases. Specifically, in analogy with the so-called Hassett–Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to$D$-lattices. In particular, for the$19$-dimensional case, we conjecturally obtain a continuous one-parameter interpolation between the GIT and Baily–Borel compactifications for the moduli of degree-$4$ $K3$surfaces. The analogous$18$-dimensional case, which corresponds to hyperelliptic degree-$4$ $K3$surfaces, can be verified by means of Variation of Geometric Invariant Theory (VGIT) quotients.

We present a Geometric Invariant Theory (GIT) construction which allows us to construct good projective degenerations of Hilbert schemes of points for simple degenerations. A comparison with the construction of Li and Wu shows that our GIT stack and the stack they construct are isomorphic, as are the associated coarse moduli schemes. Our construction is sufficiently explicit to obtain good control over the geometry of the singular fibres. We illustrate this by giving a concrete description of degenerations of degree $n$ Hilbert schemes of a simple degeneration with two components.

In this paper we compute the Poincare polynomial of a geometric invariant theory (GIT) quotient of the pointed linear sigma quotient. The pointed linear sigma model is a compactification of the space of degree-d, n-pointed maps from \({\mathbb P^1 \to \mathbb P^r}\) , and carries a natural action of \({G=SL_2(\mathbb C)}\) . Taking a GIT quotient under certain assumptions of n, r, d gives a projective moduli space M and we find its Poincare polynomial. This M is birational to \({{\overline{M}_{0,n}(\mathbb P^r,d)}}\) and we use this fact to find the Betti numbers of certain stable map spaces.

We give a geometric invariant theory (GIT) construction of the log canonical model M ¯ g (a) of the pairs (M ¯ g ,ad) for a?(7/10�?,7/10] for small ??Q + . We show that M ¯ g (7/10) is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; M ¯ g (7/10-?) is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction ?:M ¯ g (7/10+?)?M ¯ g (7/10) that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction ? + :M ¯ g (7/10-?)?M ¯ g (7/10) that is the Mori flip of ? .