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Articles 1 - 6 of 6
Full-Text Articles in Algebraic Geometry
A Zariski-Nagata Theorem For Smooth Toric Surfaces, Jordan Vincent Barrett
A Zariski-Nagata Theorem For Smooth Toric Surfaces, Jordan Vincent Barrett
Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–
The Affine Zariski-Nagata theorem is a classical result in commutative algebra that gives an expression for the nth symbolic power of a radical ideal I in a polynomial ring over a field in terms of the nth ordinary powers of the maximal ideals in [affine variety]max(I). In this thesis we discuss a well-known projective analog of Zariski-Nagata and provide the necessary background on toric varieties to present a generalization of this result to toric surfaces. We conclude with a brief discussion about work toward characterizing which abstract toric varieties have smooth point fibers with …
Betti Numbers Of Generic Ideals, Jason R. Howell
Betti Numbers Of Generic Ideals, Jason R. Howell
Electronic Theses & Dissertations (2024 - present)
We present results related to Betti numbers of so called generic ideals over a polynomial ring $T = \Bbbk[x_1, \ldots, x_n]$. For each $m$ define $T(m) = T/(x_{m+1}, \ldots, x_n)$ and for a homogeneous ideal $J$ we use the notation $J(m) = JT(m)$. Also we set $Q(m) = T(m)/J(m)$, and $L(m) = ann_{Q(m)}(x_m)$. The first main result is Theorem \ref{Long Exact Sequence} where we produce the following long exact sequence \begin{align*} \cdots \rightarrow &Tor_{k+1,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-1,j+1}^{T(m-1)}(L(m),\Bbbk)_{j-1} \rightarrow Tor_{k,j}^{T(m)}(Q(m),\Bbbk) \rightarrow \\ &Tor_{k,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-2,j-1}^{T(m-1)}(L(m),\Bbbk) \rightarrow \cdots. \end{align*} The second main result is Theorem \ref{Theorem F(j,m) equiv k(j,m)} where we explicitly describe …
Unexpectedness Stratified By Codimension, Frank Zimmitti
Unexpectedness Stratified By Codimension, Frank Zimmitti
Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–
A recent series of papers, starting with the paper of Cook, Harbourne, Migliore and Nagel on the projective plane in 2018, studies a notion of unexpectedness for finite sets Z of points in N-dimensional projective space. Say the complete linear system L of forms of degree d vanishing on Z has dimension t yet for any general point P the linear system of forms vanishing on Z with multiplicity m at P is nonempty. If the dimension of L is more than the expected dimension of t−r, where r is N+m−1 choose N …
Symbolic Powers Of Edge Ideals, Mike Janssen
Symbolic Powers Of Edge Ideals, Mike Janssen
Faculty Work Comprehensive List
No abstract provided.
Closure Operations In Commutative Rings, Chloette Joy Samsam
Closure Operations In Commutative Rings, Chloette Joy Samsam
Theses Digitization Project
The purpose of this study is to survey different types of closures and closure operations on commutative rings and ideals.
Primary Decomposition Of Ideals In A Ring, Sola Oyinsan
Primary Decomposition Of Ideals In A Ring, Sola Oyinsan
Theses Digitization Project
The concept of unique factorization was first recognized in the 1840s, but even then, it was still fairly believed to be automatic. The error of this assumption was exposed largely through attempts to prove Pierre de Fermat's, 1601-1665, last theorem. Once mathematicians discovered that this property did not always hold, it was only natural for them to try to search for the strongest available alternative. Thus began the attempt to generalize unique factorization. Using the ascending chain condition on principle ideals, we will show the conditions under which a ring is a unique factorization domain.