{"id":{"repo_id":"tuebingen","oai_identifier":"oai:publikationen.uni-tuebingen.de:10900/49066"},"canonical_url":"https://search.dev.ndltd.org/etd/tuebingen/oai:publikationen.uni-tuebingen.de:10900/49066","repository":{"repo_id":"tuebingen","name":"Universität Tübingen","base_url":"https://publikationen.uni-tuebingen.de/oai/request"},"display":{"title":"McKay correspondence and G-Hilbert schemes","abstract":"The observation of McKay relates exceptional curves in the minimal resolution of quotient singularities $\\A^2_\\C/G$ for finite subgroups $G\\subset\\SL(2,\\C)$ to the representation theory of the group $G$: the representation graph of the group $G$ is isomorphic to the intersection graph of the exceptional curves, both are graphs of ADE type (classical McKay correspondence). The McKay correspondence in a broader sense describes the geometry of resolutions of quotient singularities $X/G$ in terms of the $G$-equivariant geometry of $X$. A method to construct resolutions of quotient singularities is the $G$-Hilbert scheme $\\GHilb X$ for a scheme $X$ with $G$-operation. It parametrises $G$-clusters, these are $G$-stable finite closed subschemes $Z\\sub X$, whose coordinate ring as a representation is isomorphic to the regular representation of $G$. In this work we consider McKay correspondence over fields that are not necessarily algebraically closed and for finite group schemes instead of simply finite groups. Let $G\\subset\\SL(2,K)$ be a finite subgroup scheme over a field $K$ of characteristic $0$. Over non algebraically closed $K$ there may exist both representations of $G$ and components of the exceptional divisor in the minimal resolution of $\\A^2_K/G$ that are irreducible over $K$ but split over the algebraic closure. We show that these two kinds of splittings that arise by extending the ground field are linked and formulate a McKay correspondence relating nontrivial irreducible representations to exceptional prime divisors over arbitrary fields $K$ of characteristic $0$. With the aim to generalise the McKay correspondence, we generalise the $G$-Hilbert scheme construction to finite group schemes. Further, we introduce relative $G$-Hilbert schemes associated to a scheme with $G$-operation over another scheme and vary the base scheme. This allows to construct the $G$-Hilbert scheme without using the Hilbert scheme of $n$ points. This new construction works under more natural hypotheses, moreover, it yields additional information about the morphism from the $G$-Hilbert scheme to the quotient, which is interpreted as the structure morphism of a relative $G$-Hilbert scheme.","abstract_html":"The observation of McKay relates exceptional curves in the minimal resolution of quotient singularities <span class=\"etd-inline-math\">\\A<sup>2</sup><sub>\\</sub>C/G</span> for finite subgroups $G\\subset\\SL(2,\\C)$ to the representation theory of the group $G$: the representation graph of the group $G$ is isomorphic to the intersection graph of the exceptional curves, both are graphs of ADE type (classical McKay correspondence). The McKay correspondence in a broader sense describes the geometry of resolutions of quotient singularities $X/G$ in terms of the $G$-equivariant geometry of $X$. A method to construct resolutions of quotient singularities is the $G$-Hilbert scheme $\\GHilb X$ for a scheme $X$ with $G$-operation. It parametrises $G$-clusters, these are $G$-stable finite closed subschemes $Z\\sub X$, whose coordinate ring as a representation is isomorphic to the regular representation of $G$. In this work we consider McKay correspondence over fields that are not necessarily algebraically closed and for finite group schemes instead of simply finite groups. Let $G\\subset\\SL(2,K)$ be a finite subgroup scheme over a field $K$ of characteristic $0$. Over non algebraically closed $K$ there may exist both representations of $G$ and components of the exceptional divisor in the minimal resolution of <span class=\"etd-inline-math\">\\A<sup>2</sup><sub>K</sub>/G</span> that are irreducible over $K$ but split over the algebraic closure. We show that these two kinds of splittings that arise by extending the ground field are linked and formulate a McKay correspondence relating nontrivial irreducible representations to exceptional prime divisors over arbitrary fields $K$ of characteristic $0$. With the aim to generalise the McKay correspondence, we generalise the $G$-Hilbert scheme construction to finite group schemes. Further, we introduce relative $G$-Hilbert schemes associated to a scheme with $G$-operation over another scheme and vary the base scheme. This allows to construct the $G$-Hilbert scheme without using the Hilbert scheme of $n$ points. This new construction works under more natural hypotheses, moreover, it yields additional information about the morphism from the $G$-Hilbert scheme to the quotient, which is interpreted as the structure morphism of a relative $G$-Hilbert scheme.","abstract_has_math":true,"creators":["Blume, Mark"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2007,"date_issued":"2007","date_published":"2007","updated_at":"2026-08-21T22:21:56Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["hdl:10900/49066"],"render_values":[{"text":"hdl:10900/49066","href":null,"code":true}]}]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"source_record":{"url":"https://publikationen.uni-tuebingen.de/oai/request?verb=GetRecord&metadataPrefix=mets&identifier=oai%3Apublikationen.uni-tuebingen.de%3A10900%2F49066","prefix":"mets"},"metadata_groups":[{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2007"]},{"key":"dc:type","label":"Dc Type","values":["PhDThesis"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["hdl:10900/49066"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.other","label":"Dc Description Other","values":["The observation of McKay relates exceptional curves in the minimal resolution of quotient singularities $\\A^2_\\C/G$ for finite subgroups $G\\subset\\SL(2,\\C)$ to the representation theory of the group $G$: the representation graph of the group $G$ is isomorphic to the intersection graph of the exceptional curves, both are graphs of ADE type (classical McKay correspondence). The McKay correspondence in a broader sense describes the geometry of resolutions of quotient singularities $X/G$ in terms of the $G$-equivariant geometry of $X$. A method to construct resolutions of quotient singularities is the $G$-Hilbert scheme $\\GHilb X$ for a scheme $X$ with $G$-operation. It parametrises $G$-clusters, these are $G$-stable finite closed subschemes $Z\\sub X$, whose coordinate ring as a representation is isomorphic to the regular representation of $G$. In this work we consider McKay correspondence over fields that are not necessarily algebraically closed and for finite group schemes instead of simply finite groups. Let $G\\subset\\SL(2,K)$ be a finite subgroup scheme over a field $K$ of characteristic $0$. Over non algebraically closed $K$ there may exist both representations of $G$ and components of the exceptional divisor in the minimal resolution of $\\A^2_K/G$ that are irreducible over $K$ but split over the algebraic closure. We show that these two kinds of splittings that arise by extending the ground field are linked and formulate a McKay correspondence relating nontrivial irreducible representations to exceptional prime divisors over arbitrary fields $K$ of characteristic $0$. With the aim to generalise the McKay correspondence, we generalise the $G$-Hilbert scheme construction to finite group schemes. Further, we introduce relative $G$-Hilbert schemes associated to a scheme with $G$-operation over another scheme and vary the base scheme. This allows to construct the $G$-Hilbert scheme without using the Hilbert scheme of $n$ points. This new construction works under more natural hypotheses, moreover, it yields additional information about the morphism from the $G$-Hilbert scheme to the quotient, which is interpreted as the structure morphism of a relative $G$-Hilbert scheme.","Die Beobachtung von McKay setzt exzeptionelle Kurven in der minimalen Auflösung von Quotientensingularitäten $\\A^2_\\C/G$ für endliche Untergruppen $G\\subset\\SL(2,\\C)$ in Beziehung zu der Darstellungstheorie der Gruppe $G$: Der Darstellungsgraph der Gruppe $G$ ist isomorph zum Schnittgraph der exzeptionellen Kurven, beides sind Graphen vom ADE-Typ (klassische McKay-Korrespondenz). Die McKay-Korrespondenz im weiteren Sinne beschreibt die Geometrie von Auflösungen von Quotientensingularitäten $X/G$ mittels der $G$-äquivarianten Geometrie von $X$. Eine Methode, Auflösungen von Quotientensingularitäten zu konstruieren, ist das $G$-Hilbertschema $\\GHilb X$ zu einem Schema $X$ mit $G$-Operation. Es parametrisiert $G$-Cluster, dies sind $G$-stabile endliche abgeschlossene Unterschemata $Z\\sub X$, deren Koordinatenring als Darstellung isomorph zu der regulären Darstellung von $G$ ist. In dieser Arbeit betrachten wir McKay-Korrespondenz über nicht notwendig algebraisch abgeschlossenen Grundkörpern und für endliche Gruppenschemata statt lediglich für endliche Gruppen. Sei $G\\subset\\SL(2,K)$ ein endliches Untergruppenschema über einem Körper $K$ der Charakteristik $0$. Über nicht algebraisch abgeschlossenem $K$ kann es sowohl Darstellungen von $G$ als auch Komponenten des exzeptionellen Divisors in der minimalen Auflösung von $\\A^2_K/G$ geben, die irreduzibel über $K$ sind, aber über dem algebraischen Abschluss zerfallen. Wir zeigen, dass diese beiden Arten, bei Erweiterung des Grundkörpers zu zerfallen, miteinander verbunden sind, und formulieren eine McKay-Korrespondenz, die nichttriviale irreduzible Darstellungen mit exzeptionellen Primdivisoren in Beziehung setzt, für beliebige Körper $K$ der Charakteristik $0$. Mit dem Ziel, die McKay-Korrespondenz zu verallgemeinern, verallgemeinern wir die Konstruktion von $G$-Hilbertschemata auf endliche Gruppenschemata. Weiter führen wir relative $G$-Hilbertschemata zu einem Schema mit $G$-Operation über einem anderen Schema ein und variieren das Basisschema. Dies erlaubt es, das $G$-Hilbertschema ohne Verwendung des Hilbertschemas von $n$ Punkten zu konstruieren. Diese neue Konstruktion funktioniert unter natürlicheren Voraussetzungen, darüber hinaus liefert sie zusätzliche Informationen über den Morphismus vom $G$-Hilbertschema auf den Quotienten, dieser wird interpretiert als Strukturmorphismus eines relativen $G$-Hilbertschemas."]},{"key":"dc:title","label":"Title","values":["McKay correspondence and G-Hilbert schemes"]}]}],"canonical_facts":{"dc:date.issued":["2007"],"dc:description.other":["The observation of McKay relates exceptional curves in the minimal resolution of quotient singularities $\\A^2_\\C/G$ for finite subgroups $G\\subset\\SL(2,\\C)$ to the representation theory of the group $G$: the representation graph of the group $G$ is isomorphic to the intersection graph of the exceptional curves, both are graphs of ADE type (classical McKay correspondence). The McKay correspondence in a broader sense describes the geometry of resolutions of quotient singularities $X/G$ in terms of the $G$-equivariant geometry of $X$. A method to construct resolutions of quotient singularities is the $G$-Hilbert scheme $\\GHilb X$ for a scheme $X$ with $G$-operation. It parametrises $G$-clusters, these are $G$-stable finite closed subschemes $Z\\sub X$, whose coordinate ring as a representation is isomorphic to the regular representation of $G$. In this work we consider McKay correspondence over fields that are not necessarily algebraically closed and for finite group schemes instead of simply finite groups. Let $G\\subset\\SL(2,K)$ be a finite subgroup scheme over a field $K$ of characteristic $0$. Over non algebraically closed $K$ there may exist both representations of $G$ and components of the exceptional divisor in the minimal resolution of $\\A^2_K/G$ that are irreducible over $K$ but split over the algebraic closure. We show that these two kinds of splittings that arise by extending the ground field are linked and formulate a McKay correspondence relating nontrivial irreducible representations to exceptional prime divisors over arbitrary fields $K$ of characteristic $0$. With the aim to generalise the McKay correspondence, we generalise the $G$-Hilbert scheme construction to finite group schemes. Further, we introduce relative $G$-Hilbert schemes associated to a scheme with $G$-operation over another scheme and vary the base scheme. This allows to construct the $G$-Hilbert scheme without using the Hilbert scheme of $n$ points. This new construction works under more natural hypotheses, moreover, it yields additional information about the morphism from the $G$-Hilbert scheme to the quotient, which is interpreted as the structure morphism of a relative $G$-Hilbert scheme.","Die Beobachtung von McKay setzt exzeptionelle Kurven in der minimalen Auflösung von Quotientensingularitäten $\\A^2_\\C/G$ für endliche Untergruppen $G\\subset\\SL(2,\\C)$ in Beziehung zu der Darstellungstheorie der Gruppe $G$: Der Darstellungsgraph der Gruppe $G$ ist isomorph zum Schnittgraph der exzeptionellen Kurven, beides sind Graphen vom ADE-Typ (klassische McKay-Korrespondenz). Die McKay-Korrespondenz im weiteren Sinne beschreibt die Geometrie von Auflösungen von Quotientensingularitäten $X/G$ mittels der $G$-äquivarianten Geometrie von $X$. Eine Methode, Auflösungen von Quotientensingularitäten zu konstruieren, ist das $G$-Hilbertschema $\\GHilb X$ zu einem Schema $X$ mit $G$-Operation. Es parametrisiert $G$-Cluster, dies sind $G$-stabile endliche abgeschlossene Unterschemata $Z\\sub X$, deren Koordinatenring als Darstellung isomorph zu der regulären Darstellung von $G$ ist. In dieser Arbeit betrachten wir McKay-Korrespondenz über nicht notwendig algebraisch abgeschlossenen Grundkörpern und für endliche Gruppenschemata statt lediglich für endliche Gruppen. Sei $G\\subset\\SL(2,K)$ ein endliches Untergruppenschema über einem Körper $K$ der Charakteristik $0$. Über nicht algebraisch abgeschlossenem $K$ kann es sowohl Darstellungen von $G$ als auch Komponenten des exzeptionellen Divisors in der minimalen Auflösung von $\\A^2_K/G$ geben, die irreduzibel über $K$ sind, aber über dem algebraischen Abschluss zerfallen. Wir zeigen, dass diese beiden Arten, bei Erweiterung des Grundkörpers zu zerfallen, miteinander verbunden sind, und formulieren eine McKay-Korrespondenz, die nichttriviale irreduzible Darstellungen mit exzeptionellen Primdivisoren in Beziehung setzt, für beliebige Körper $K$ der Charakteristik $0$. Mit dem Ziel, die McKay-Korrespondenz zu verallgemeinern, verallgemeinern wir die Konstruktion von $G$-Hilbertschemata auf endliche Gruppenschemata. Weiter führen wir relative $G$-Hilbertschemata zu einem Schema mit $G$-Operation über einem anderen Schema ein und variieren das Basisschema. Dies erlaubt es, das $G$-Hilbertschema ohne Verwendung des Hilbertschemas von $n$ Punkten zu konstruieren. Diese neue Konstruktion funktioniert unter natürlicheren Voraussetzungen, darüber hinaus liefert sie zusätzliche Informationen über den Morphismus vom $G$-Hilbertschema auf den Quotienten, dieser wird interpretiert als Strukturmorphismus eines relativen $G$-Hilbertschemas."],"dc:identifier":["hdl:10900/49066"],"dc:title":["McKay correspondence and G-Hilbert schemes"],"dc:type":["PhDThesis"]},"updated_at":"2026-08-21T22:21:56Z"}