TY  - JOUR
T1  - The one-dimensional nonreduced line scheme of two families of quadratic quantum $\mathbb{P}^{\bf 3}$s
AU  - Vancliff, Michaela
AU  - Lim, Ian C.
AU  - Lozano, Jose E.
AU  - Mastriania Iv, Anthony
PY  - 2025
DA  - February
Y2  - 2024
DO  - 10.24330/ieja.1643942
JF  - International Electronic Journal of Algebra
JO  - IEJA
PB  - Abdullah HARMANCI
WT  - DergiPark
SN  - 1306-6048
SP  - 1
EP  - 20
LA  - en
AB  - The classification of quantum $\mathbb{P}^2$s was completed by M. Artin et al. decades ago, but the classification of quadratic algebras that are viewed as quantum $\mathbb{P}^3$s is still an open problem. Based on work of M. Van den Bergh, it is believed that a ``generic'' quadratic quantum $\mathbb{P}^3$ should have a finite point scheme and a one-dimensional line scheme. Two families of quadratic quantum $\mathbb{P}^3$s with these geometric properties are presented herein, where each family member has a line scheme that is either a union of lines or is a union of a line, a conic and a curve.  Moreover, we prove that, under certain conditions, if $A$ is a quadratic quantum $\mathbb{P}^3$ that contains a subalgebra $B$ that is a quadratic quantum $\mathbb{P}^2$, then the point scheme of $B$ embeds in the line scheme of $A$.
KW  - Regular algebra
KW  - quadratic algebra
KW  - Ore extension
KW  - point module
KW  - line module
KW  - point scheme
KW  - line scheme
KW  - quantum
CR  - M. Artin and W. Schelter, Graded algebras of global dimension $3$, Adv. in Math., 66(2) (1987), 171-216.
CR  - M. Artin,  J. Tate and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, in The Grothendieck Festschrift 1, Eds. P. Cartier et al., Progr. Math., 86 (1990), 33-85.
CR  - M. Artin, J. Tate and M. Van den Bergh, Modules over regular algebras of dimension $3$, Invent. Math., 106(2) (1991), 335-388.
CR  - T. Cassidy and M. Vancliff, Generalizations of graded Clifford algebras and of complete intersections, J. Lond. Math. Soc. (2), 81(1) (2010), 91-112. (Corrigendum: 90(2) (2014), 631-636.)
CR  - R. G. Chandler and M. Vancliff, The one-dimensional line scheme of a certain family of quantum $\mathbb{P}^3$s, J. Algebra, 439 (2015), 316-333.
CR  - A. Chirvasitu and S. P. Smith, Exotic elliptic algebras of dimension $4$, with an appendix by D. Tomlin, Adv. Math., 309 (2017), 558-623.
CR  - A. Chirvasitu, S. P. Smith and M. Vancliff, A geometric invariant of $6$-dimensional subspaces of $4 \times 4$ matrices, Proc. Amer. Math.
Soc., 148(3) (2020), 915-928.
CR  - W. Decker, G.-M. Greuel, G. Pfister and H. Schonemann, SINGULAR 4-4-0 - A Computer Algebra System for Polynomial Computations, 2024: https://www.singular.uni-kl.de.
CR  - P. Goetz, E. Kirkman, W. F. Moore and K. B. Vashaw, Some Artin-Schelter regular algebras from dual reflection groups and their geometry, (2024), arXiv:2410.08959v1 [math.RA].
CR  - P. Goetz and B. Shelton, Representation theory of two families of quantum projective $3$-spaces, J. Algebra, 295(1) (2006), 141-156.
CR  - T. Levasseur, Some properties of non-commutative regular graded rings, Glasgow Math. J., 34(3) (1992), 277-300.
CR  - T. Levasseur and S. P. Smith, Modules over the $4$-dimensional Sklyanin algebra, Bull. Soc. Math. France, 121(1) (1993), 35-90.
CR  - I. C. Lim,  Some Quadratic Quantum $\mathbb{P}^3$s with a Linear One-Dimensional Line Scheme, Ph.D. Dissertation, Univ. of Texas at Arlington, 2021: https://mavmatrix.uta.edu/math\_dissertations/174/.
CR  - J. E. Lozano, Point Modules and Line Modules of Certain Quadratic Quantum Projective Spaces, Ph.D. Dissertation, Univ. of Texas at Arlington, 2024: https://mavmatrix.uta.edu/math\_dissertations/2/.
CR  - A. Mastriania IV, Some Quadratic Regular Algebras on Four Generators with a 1-Dimensional Nonreduced Line Scheme, Ph.D. Dissertation, 
Univ. of Texas at Arlington, 2019: https://mavmatrix.uta.edu/math\_dissertations/214/.
CR  - Mathematica, Version 12.0, Wolfram Research Inc., Champaign, IL, 2019.
CR  - Maxima, Version 5.47.0 - A Computer Algebra System, 2023: https://maxima.sourceforge.io/.
CR  - B. Shelton and M. Vancliff, Schemes of line modules I, J. London Math. Soc. (2), 65(3) (2002), 575-590.
CR  - B. Shelton and M. Vancliff, Schemes of line modules II, Comm. Algebra, 30(5) (2002), 2535-2552.
CR  - D. R. Stephenson, Artin-Schelter regular algebras of global dimension three, J. Algebra, 183(1) (1996), 55-73.
CR  - D. R. Stephenson, Algebras associated to elliptic curves, Trans. Amer. Math. Soc., 349(6) (1997), 2317-2340.
CR  - D. R. Stephenson, Quantum planes of weight $(1, 1, n)$, J. Algebra, 225(1) (2000), 70-92.
CR  - D. R. Stephenson and M. Vancliff, Some finite quantum $\mathbb{P}^3$s that are infinite modules over their centers, J. Algebra, 297(1) (2006), 208-215.
CR  - D. Tomlin and M. Vancliff, The one-dimensional line scheme of a family of quadratic quantum $\mathbb{P}^3$s, J. Algebra, 502 (2018), 588-609.
CR  - M. Vancliff, The interplay of algebra ant geometry in the setting of regular algebras, in Commutative Algebra and Noncommutative Algebraic Geometry, Eds. D. Eisenbud et al., Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, New York, 67 (2015), 371-390.
UR  - https://doi.org/10.24330/ieja.1643942
L1  - https://dergipark.org.tr/en/download/article-file/4627754
ER  -