Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Higher-dimensional analogs of Châtelet surfaces

Higher-dimensional analogs of Châtelet surfaces (Bull. London Math. Soc. 44 (2012) 125–135) Let $$K/k$$ be a cyclic Galois extension of fields of degree $$n$$, and let $$P(x) \in k[x]$$ be a separable polynomial of degree $$dn$$ or $$dn - 1$$. Let $$X_0$$ be the affine norm hypersurface in $${{\mathbb A}}^{n+1}_k$$ given by   \[{{\mathrm {{\textit {N}}}}}_{K/k}(\vec {z}) = P(x) \neq 0.\] (1) In [1, §2], we attempted to construct a smooth proper model $$X$$ of $$X_0$$ extending the map $$X_0 \to {{\mathbb A}}^1_k\setminus V(P(x))$$ given by $$(\vec {z},x) \mapsto x$$ to a map $$X \to {{\mathbb P}}^1_k$$. However, [1, Proposition 2.1] is false whenever $$n >2$$. In this note, we explain how all statements and proofs of [1] can be rectified using a construction from [2], particularly the following theorem. Theorem 1 [2, Theorem 1] Let$$K/k$$be a cyclic Galois extension of fields of degree$$n,$$and let$$P(x) \in k[x]$$be a separable polynomial of degree$$dn$$or$$dn - 1$$. There exists a smooth proper compactification$$X$$of$$X_0,$$fibered over$${{\mathbb P}}^1_k = {{\mathrm {Proj}}}k[x_0,x_1],$$such that$$X \to {{\mathbb P}}^1_k$$extends the map$$X_0 \to {{\mathbb A}}^1_k$$. Furthermore, the generic fiber of$$X \to {{\mathbb P}}^1_k$$is a Severi–Brauer variety, and the degenerate fibers lie over$$V(P(x_0/x_1)x_1^{dn}),$$and consist of the union of$$n$$rational varieties all conjugate under$${{\mathrm {Gal}}}(K/k)$$. Replacing the variety $$X$$ of [1, §2] with the variety $$X$$ from Theorem 1 immediately rectifies all but one of the statements and proofs of [1, §§3–5] (we return to this exception below). More precisely, the proofs of [1, Proposition 3.1 and Theorem 3.2] depend only on That these properties hold is exactly the result of Theorem 1. The remaining proofs of statements in [1, §§4–5] rely on [1, Proposition 3.1 and Theorem 3.2] without further mention of the specific geometry of the fibration $$X\to {{\mathbb P}}^1$$, except for part of the proof of [1, Theorem 1.3]. the generic fiber of $$X \to {{\mathbb P}}^1$$ being a Severi–Brauer variety and the degenerate fibers of $$X\to {{\mathbb P}}^1$$ lying over $$V(P(x_0/x_1)x_1^{dn})$$ and consisting of the union of $$n$$ rational varieties, all conjugate under $${{\mathrm {Gal}}}(K/k)$$. To rectify the remaining part of the proof of [1, Theorem 1.3], we must correct the construction of the Châtelet $$p$$-fold bundle over $${{\mathbb P}}^1\times {{\mathbb P}}^1$$ given by $$u^pP_{\infty }(x) +P_0(x)$$. To do so, we carry out the same constructions as in [2] over the polynomial rings $$k[u,x]$$, $$k[u^{-1},x]$$, $$k[u,x^{-1}]$$, and $$k[u^{-1},x^{-1}]$$ and glue to construct a bundle over $${{\mathbb P}}^1\times {{\mathbb P}}^1$$. More precisely, the proof of [1, Theorem 1.3] requires a smooth compactification of the normic bundle $$X_0 \to U\subset {{\mathbb A}}^2$$ given by   \[{{\mathrm {{\textit {N}}}}}_{K/k}(\vec {z}) = u^pP_{\infty }(x) +P_0(x) \neq 0,\] where the closure of $$V(u^pP_{\infty }(x) +P_0(x))$$ in $${{\mathbb P}}^1_u\times {{\mathbb P}}^1_x$$ is smooth and of bidegree $$(d_1p,d_2p)$$ for some positive integers $$d_1$$ and $$d_2$$. (In characteristic $$p$$, we instead consider the normic bundle $$X_0$$ given by $${{\mathrm {{\textit {N}}}}}_{K/k}(\vec {z}) = u^pP_{\infty }(x) +u^{p-1}P_{\infty }(x) +P_0(x) \neq 0$$, with the same conditions on the closure of the curve in $${{\mathbb P}}^1_u\times {{\mathbb P}}^1_x$$.) Sections 3 and 4 of [2] give a smooth partial compactification $$X\to {{\mathrm {Spec}}}R$$ of any normic bundle $$X_0\to D(a)\subset {{\mathrm {Spec}}}R$$ given by $$N_{K/k}(\vec {z}) = a\neq 0$$ for any $$k$$-algebra $$R$$, as long as $$V(a)$$ is smooth in $${{\mathrm {Spec}}}R$$. Thus, we may apply [2, §§3 and 4] to construct relative compactifications over the polynomial rings $$k[u,x]$$, $$k[u^{-1},x]$$, $$k[u,x^{-1}]$$, and $$k[u^{-1},x^{-1}]$$. Then, since the closure of $$V(u^pP_{\infty }(x) +P_0(x))$$ (respectively, $$V(u^pP_{\infty }(x) +u^{p-1}P_{\infty }(x) +P_0(x))$$) has bidegree $$(d_1p,d_2p)$$ for some positive integers $$d_1$$ and $$d_2$$, we may glue the relative compactifications as in [2, Lemma 4] to construct a smooth proper model $$X\to {{\mathbb P}}^1\times {{\mathbb P}}^1$$. References 1 Várilly-Alvarado A. Viray B., ‘ Higher-dimensional analogs of Châtelet surfaces’, Bull. London Math. Soc.  44 ( 2012) 125– 135. Google Scholar CrossRef Search ADS   2 Várilly-Alvarado A. Viray B., ‘ Smooth compactifications of certain normic bundles’, Eur. J. Math. , to appear, https://doi.org/10.1007/s40879-015-0035-7. © 2015 London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Higher-dimensional analogs of Châtelet surfaces

Download PDF
2 pages

Loading next page...
 
/lp/oxford-university-press/higher-dimensional-analogs-of-ch-telet-surfaces-AgKCrPyTLM

References (2)

Publisher
Oxford University Press
Copyright
© 2015 London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/bdv003
Publisher site
See Article on Publisher Site

Abstract

(Bull. London Math. Soc. 44 (2012) 125–135) Let $$K/k$$ be a cyclic Galois extension of fields of degree $$n$$, and let $$P(x) \in k[x]$$ be a separable polynomial of degree $$dn$$ or $$dn - 1$$. Let $$X_0$$ be the affine norm hypersurface in $${{\mathbb A}}^{n+1}_k$$ given by   \[{{\mathrm {{\textit {N}}}}}_{K/k}(\vec {z}) = P(x) \neq 0.\] (1) In [1, §2], we attempted to construct a smooth proper model $$X$$ of $$X_0$$ extending the map $$X_0 \to {{\mathbb A}}^1_k\setminus V(P(x))$$ given by $$(\vec {z},x) \mapsto x$$ to a map $$X \to {{\mathbb P}}^1_k$$. However, [1, Proposition 2.1] is false whenever $$n >2$$. In this note, we explain how all statements and proofs of [1] can be rectified using a construction from [2], particularly the following theorem. Theorem 1 [2, Theorem 1] Let$$K/k$$be a cyclic Galois extension of fields of degree$$n,$$and let$$P(x) \in k[x]$$be a separable polynomial of degree$$dn$$or$$dn - 1$$. There exists a smooth proper compactification$$X$$of$$X_0,$$fibered over$${{\mathbb P}}^1_k = {{\mathrm {Proj}}}k[x_0,x_1],$$such that$$X \to {{\mathbb P}}^1_k$$extends the map$$X_0 \to {{\mathbb A}}^1_k$$. Furthermore, the generic fiber of$$X \to {{\mathbb P}}^1_k$$is a Severi–Brauer variety, and the degenerate fibers lie over$$V(P(x_0/x_1)x_1^{dn}),$$and consist of the union of$$n$$rational varieties all conjugate under$${{\mathrm {Gal}}}(K/k)$$. Replacing the variety $$X$$ of [1, §2] with the variety $$X$$ from Theorem 1 immediately rectifies all but one of the statements and proofs of [1, §§3–5] (we return to this exception below). More precisely, the proofs of [1, Proposition 3.1 and Theorem 3.2] depend only on That these properties hold is exactly the result of Theorem 1. The remaining proofs of statements in [1, §§4–5] rely on [1, Proposition 3.1 and Theorem 3.2] without further mention of the specific geometry of the fibration $$X\to {{\mathbb P}}^1$$, except for part of the proof of [1, Theorem 1.3]. the generic fiber of $$X \to {{\mathbb P}}^1$$ being a Severi–Brauer variety and the degenerate fibers of $$X\to {{\mathbb P}}^1$$ lying over $$V(P(x_0/x_1)x_1^{dn})$$ and consisting of the union of $$n$$ rational varieties, all conjugate under $${{\mathrm {Gal}}}(K/k)$$. To rectify the remaining part of the proof of [1, Theorem 1.3], we must correct the construction of the Châtelet $$p$$-fold bundle over $${{\mathbb P}}^1\times {{\mathbb P}}^1$$ given by $$u^pP_{\infty }(x) +P_0(x)$$. To do so, we carry out the same constructions as in [2] over the polynomial rings $$k[u,x]$$, $$k[u^{-1},x]$$, $$k[u,x^{-1}]$$, and $$k[u^{-1},x^{-1}]$$ and glue to construct a bundle over $${{\mathbb P}}^1\times {{\mathbb P}}^1$$. More precisely, the proof of [1, Theorem 1.3] requires a smooth compactification of the normic bundle $$X_0 \to U\subset {{\mathbb A}}^2$$ given by   \[{{\mathrm {{\textit {N}}}}}_{K/k}(\vec {z}) = u^pP_{\infty }(x) +P_0(x) \neq 0,\] where the closure of $$V(u^pP_{\infty }(x) +P_0(x))$$ in $${{\mathbb P}}^1_u\times {{\mathbb P}}^1_x$$ is smooth and of bidegree $$(d_1p,d_2p)$$ for some positive integers $$d_1$$ and $$d_2$$. (In characteristic $$p$$, we instead consider the normic bundle $$X_0$$ given by $${{\mathrm {{\textit {N}}}}}_{K/k}(\vec {z}) = u^pP_{\infty }(x) +u^{p-1}P_{\infty }(x) +P_0(x) \neq 0$$, with the same conditions on the closure of the curve in $${{\mathbb P}}^1_u\times {{\mathbb P}}^1_x$$.) Sections 3 and 4 of [2] give a smooth partial compactification $$X\to {{\mathrm {Spec}}}R$$ of any normic bundle $$X_0\to D(a)\subset {{\mathrm {Spec}}}R$$ given by $$N_{K/k}(\vec {z}) = a\neq 0$$ for any $$k$$-algebra $$R$$, as long as $$V(a)$$ is smooth in $${{\mathrm {Spec}}}R$$. Thus, we may apply [2, §§3 and 4] to construct relative compactifications over the polynomial rings $$k[u,x]$$, $$k[u^{-1},x]$$, $$k[u,x^{-1}]$$, and $$k[u^{-1},x^{-1}]$$. Then, since the closure of $$V(u^pP_{\infty }(x) +P_0(x))$$ (respectively, $$V(u^pP_{\infty }(x) +u^{p-1}P_{\infty }(x) +P_0(x))$$) has bidegree $$(d_1p,d_2p)$$ for some positive integers $$d_1$$ and $$d_2$$, we may glue the relative compactifications as in [2, Lemma 4] to construct a smooth proper model $$X\to {{\mathbb P}}^1\times {{\mathbb P}}^1$$. References 1 Várilly-Alvarado A. Viray B., ‘ Higher-dimensional analogs of Châtelet surfaces’, Bull. London Math. Soc.  44 ( 2012) 125– 135. Google Scholar CrossRef Search ADS   2 Várilly-Alvarado A. Viray B., ‘ Smooth compactifications of certain normic bundles’, Eur. J. Math. , to appear, https://doi.org/10.1007/s40879-015-0035-7. © 2015 London Mathematical Society

Journal

Bulletin of the London Mathematical Society – Oxford University Press

Published: Feb 18, 2015

There are no references for this article.