Toric degenerations of projective varieties with an application to equivariant Hilbert functions

Murata, Takuya (2020) Toric degenerations of projective varieties with an application to equivariant Hilbert functions. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

A toric degeneration is a flat family over $\mathbb{A}^1$ that is trivial away from the special fiber (fiber over zero) and whose special fiber is a variety acted linearly by a torus with a dense orbit; i.e., the special fiber is a non-normal = not-necessarily-normal toric variety. We introduce a systematic method to construct toric degenerations of a projective variety (embedded up to Veronese embeddings). Part 1 develops the general theory of non-normal toric varieties by generalizing the more conventional theory of toric varieties. A new characterization of non-normal toric varieties as a complex of toric varieties is given. Given a projective variety $X$ of dimension $d$, the main result of the thesis (Part 2) constructs a finite sequence of flat degenerations with irreducible and reduced special fibers such that the last one is a non-normal toric variety. The degeneration sequence depends on the choice of a full flag of closed subvarieties $X = Y_0 \supset Y_1 \supset \cdots \supset Y_d$ such that each $Y_i$ is a \emph{good divisor} in $Y_{i-1}$. The notion of a good divisor comes from the asymptotic ideal theory in commutative algebra and the goodness ensures the finite generation of the defining graded ring of the special fiber in each step. This is a generalization of degeneration (or deformation) to normal cone in intersection theory and can be regarded as geometric reinterpretation of the construction of a valuation in \cite{Oko}, the key step in the construction of a Newton--Okounkov body. Part 3 reformulates the main result of \cite{Oko} in terms of an equivariant Hilbert function; this reformulation may be thought of as a special case of the equivariant Riemann--Roch theorem.

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Details

Item Type:	University of Pittsburgh ETD
Status:	Unpublished
Creators/Authors:	Creators Email Pitt Username ORCID Murata, Takuya takusi@gmail.com tam60 0000-0002-4539-9620
ETD Committee:	Title Member Email Address Pitt Username ORCID Committee Chair Kaveh, Kiumars kaveh@pitt.edu kaveh Committee Member Ion, Bogdan ion@pitt.edu ion Committee Member Constantine, Gregory gmc@pitt.edu gmc Committee Member Manon, Christopher chris.manon@gmail.com
Date:	16 September 2020
Date Type:	Publication
Defense Date:	14 April 2020
Approval Date:	16 September 2020
Submission Date:	1 August 2020
Access Restriction:	No restriction; Release the ETD for access worldwide immediately.
Number of Pages:	134
Institution:	University of Pittsburgh
Schools and Programs:	Dietrich School of Arts and Sciences > Mathematics
Degree:	PhD - Doctor of Philosophy
Thesis Type:	Doctoral Dissertation
Refereed:	Yes
Uncontrolled Keywords:	toric degeneration, toric variety, Newton-Okounkov body
Date Deposited:	16 Sep 2020 14:37
Last Modified:	16 Sep 2020 14:37
URI:	http://d-scholarship.pitt.edu/id/eprint/39637

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