QUADRATIC FAMILIES OF ELLIPTIC CURVES AND UNIRATIONALITY OF DEGREE 1 CONIC BUNDLES

被引：22

作者：

Kollar, Janos ^{[1
]}

Mella, Massimiliano ^{[2
]}

机构：

[1] Princeton Univ, Princeton, NJ 08544 USA

[2] Univ Ferrara, Dipartimento Matemat & Informat, Machiavelli 35, I-44100 Ferrara, Italy

来源：

AMERICAN JOURNAL OF MATHEMATICS | 2017年 / 139卷 / 04期

关键词：

DEL PEZZO SURFACES; ELEMENTS; BR-2(K(X));

D O I：

10.1353/ajm.2017.0024

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We study families of elliptic curves whose coefficients are degree 2 polynomials in a variable t. All such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers. We prove that such conic bundles are unirational. As a consequence one obtains that, for infinitely many values of t, the resulting elliptic curve has rank at least 1.

引用

页码：915 / 936

页数：22

共 27 条

[21] Mestre JF, 1996, CR ACAD SCI I-MATH, V322, P423
[22] THREEFOLDS WHOSE CANONICAL BUNDLES ARE NOT NUMERICALLY EFFECTIVE
MORI, S
[J]. ANNALS OF MATHEMATICS, 1982, 116 (01) : 133 - 176
[23] ON QUADRATIC FAMILIES OF CM ELLIPTIC CURVES
Munshi, Ritabrata
[J]. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 2011, 363 (08) : 4337 - 4358
[24] NoTHER M., 1870, Math. Ann., V3, P161, DOI [10.1007/BF01443982, DOI 10.1007/BF01443982]
[25] Ranks of elliptic curves
Rubin, K
Silverberg, A
[J]. BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 2002, 39 (04): : 455 - 474
[26] On the unirationality of del Pezzo surfaces of degree 2
Salgado, Cecilia
Testa, Damiano
Varilly-Alvarado, Anthony
[J]. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 2014, 90 : 121 - 139
[27] Segre B., 1951, MATH NOTAE, V11, P1

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