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The space of Cohen–Macaulay curves and related topics
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Algebra and Geometry)
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The space of Cohen–Macaulay curves is a compactification of the space of curves that are embedded in a given projective space Pn. The idea is similar to that of the Hilbert scheme but instead of adding degenerated curves, one considers only curves without embedded or isolated points. However, the curves need not be embedded into the projective space. Instead, they come with a finite morphism to Pn that is generically a closed immersion. More precisely, the space CM of Cohen–Macaulay curves parameterizes flat families of pairs  where  is a curve without embedded or isolated points and  is a finite morphism that is an isomorphism onto its image away from finitely many closed points and such that  has Hilbert polynomial p(t) with respect to the map .

In Paper A we show that the moduli functor CM is represented by a proper algebraic space. This is done by constructing a smooth, surjective cover  and by verifying the valuative criterion for properness.

Paper B studies the moduli space CM in the particular case n = 3 and p(t) = 3t + 1, that is, the Cohen–Macaulay space of twisted cubics. We de- scribe the points of CM and show that they are in bijection with the points on the 12-dimensional component H0 of the Hilbert scheme of twisted cu- bics. Knowing the points of CM, we can then show that the moduli space is irreducible, smooth and has dimension 12.

Paper C concerns the notion of biequidimensionality of topological spaces and Noetherian schemes. In EGA it is claimed that a topological space X is equidimensional, equicodimensional and catenary if and only if all maximal chains of irreducible closed subsets in X have the same length. We construct examples of topological spaces and Noetherian schemes showing that the sec- ond property is strictly stronger. This gives rise to two different notions of biequidimensionality, and we show how they relate to the dimension formula and the existence of a codimension function.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. , x, 18 p.
Series
TRITA-MAT-A, 2014:03
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-143966OAI: oai:DiVA.org:kth-143966DiVA: diva2:709970
Public defence
2014-04-25, D3, Lindstedtsvägen 5, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20140404

Available from: 2014-04-04 Created: 2014-04-03 Last updated: 2014-04-04Bibliographically approved
List of papers
1. The space of Cohen–Macaulay curves
Open this publication in new window or tab >>The space of Cohen–Macaulay curves
(English)Manuscript (preprint) (Other academic)
Abstract [en]

One can consider the Hilbert scheme as a natural compactification of the space of smooth projective curves with fixed Hilbert polynomial. Here we consider a different modular compactification, namely the functor CM parameterizing curves together with a finite map to  that is generically a closed immersion. We prove that CM is an algebraic space by contructing a scheme W and a representable, surjective and smooth map W -> CM. Moreover, we show that CM satisfies the valuative criterion for properness.

National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-143875 (URN)
Note

QS 2014

Available from: 2014-04-03 Created: 2014-04-01 Last updated: 2014-04-04Bibliographically approved
2. The Cohen–Macaulay space of twisted cubics
Open this publication in new window or tab >>The Cohen–Macaulay space of twisted cubics
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this work, we describe the Cohen-Macaulay space CM of twisted cubics parameterizing curves  together with a finite map  that is generically a closed immersion and such that  has Hilbert polynomial p(t)=3t+1 with respect to . We show that CM is irreducible, smooth and birational to one component of the Hilbert scheme of twisted cubics.

National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-143964 (URN)
Note

QS 2014

Available from: 2014-04-03 Created: 2014-04-03 Last updated: 2014-04-04Bibliographically approved
3. Some remarks on biequidimensionality of topological spaces and Noetherian schemes
Open this publication in new window or tab >>Some remarks on biequidimensionality of topological spaces and Noetherian schemes
(English)Manuscript (preprint) (Other academic)
Abstract [en]

There are many examples of the fact that dimension and codimension behave somewhat counterintuitively. In EGA it is stated that a topological space is equidimensional, equicodimensional and catenary if and only if every maximal chain of irreducible closed subsets has the same length. We construct examples that show that this is not even true for the spectrum of a Noetherian ring. This gives rise to two notions of biequidimensionality, and we show how these relate to the dimension formula and the existence of a codimension function.

National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-143965 (URN)
Note

QS 2014

Available from: 2014-04-03 Created: 2014-04-03 Last updated: 2014-04-04Bibliographically approved

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