scholarly journals Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $ {\mathbb P}^5$

2020 ◽  
Vol 211 (2) ◽  
pp. 161-200
Author(s):  
K. Banerjee ◽  
V. Guletskiĭ
Keyword(s):  
Rational Equivalence ◽  
Cubic Hypersurfaces

scholarly journals Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

2021 ◽  
Vol 9 ◽  
Author(s):  
Alex Chirvasitu ◽  
Ryo Kanda ◽  
S. Paul Smith
Keyword(s):  
Elliptic Curve ◽  
Symmetric Power ◽  
Canonical Homomorphism ◽  
Coordinate Ring ◽  
Characteristic Variety ◽  
Closed Subvariety ◽  
Coordinate Rings ◽  
Cubic Hypersurfaces ◽  
Elliptic Algebras ◽  
Twisted Homogeneous Coordinate Ring

Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

The Lang–Weil Estimate for Cubic Hypersurfaces

2013 ◽  
Vol 56 (3) ◽  
pp. 500-502 ◽  
Author(s):  
T. D. Browning
Keyword(s):  
Rational Points ◽  
Cubic Hypersurfaces ◽  
Weil Estimate

AbstractAn improved estimate is provided for the number of 𝔽q-rational points on a geometrically irreducible, projective, cubic hypersurface that is not equal to a cone.

Invariants of rational equivalence

2000 ◽  
Vol 41 (2) ◽  
pp. 357-361 ◽  
Author(s):  
A. G. Pinus
Keyword(s):  
Rational Equivalence

Pieri’S Formula Via Explicit Rational Equivalence

1997 ◽  
Vol 49 (6) ◽  
pp. 1281-1298 ◽  
Author(s):  
Frank Sottile
Keyword(s):  
Combinatorial Proof ◽  
Geometric Proof ◽  
Intersection Product ◽  
Rational Equivalence ◽  
Schubert Cycles

AbstractPieri’s formula describes the intersection product of a Schubert cycle by a special Schubert cycle on a Grassmannian. We present a new geometric proof, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert cycles to the intersection of a Schubert cycle with a special Schubert cycle. The geometry of these rational equivalences indicates a link to a combinatorial proof of Pieri’s formula using Schensted insertion.

scholarly journals A Mordell–Weil theorem for cubic hypersurfaces of high dimension

2017 ◽  
Vol 11 (8) ◽  
pp. 1953-1965 ◽  
Author(s):  
Stefanos Papanikolopoulos ◽  
Samir Siksek
Keyword(s):  
High Dimension ◽  
Weil Theorem ◽  
Cubic Hypersurfaces

scholarly journals On Rational Equivalence in Tropical Geometry

2016 ◽  
Vol 68 (2) ◽  
pp. 241-257 ◽  
Author(s):  
Lars Allermann ◽  
Simon Hampe ◽  
Johannes Rau
Keyword(s):  
Tropical Geometry ◽  
Chow Groups ◽  
Independent Interest ◽  
Main Step ◽  
Rational Equivalence ◽  
Basic Properties ◽  
Boundary Points ◽  
Intersection Ring ◽  
Tropical Varieties

AbstractThis article discusses the concept of rational equivalence in tropical geometry (and replaces an older, imperfect version). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the “bounded” Chow groups of Rn by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest. We show that every tropical cycle in Rn is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).

Sign in / Sign up

Export Citation Format

Share Document