RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Guidelines for authors License agreement Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Trudy MIAN: Year: Volume: Issue: Page: Find

 Tr. Mat. Inst. Steklova, 2004, Volume 246, Pages 217–239 (Mi tm157)

On Correspondences of a K3 Surface with Itself. I

V. V. Nikulinab

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of Liverpool

Abstract: Let $X$ be a K3 surface with a polarization $H$ of degree $H^2=2rs$, $r,s\ge 1$. Assume that $H\cdot N(X)=\mathbb Z$ for the Picard lattice $N(X)$. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(r,H,s)$ is again a K3 surface $Y$. We prove that $Y\cong X$ if there exists $h_1\in N(X)$ with $h_1^2=f(r,s)$, $H\cdot h_1\equiv 0\mathrm { mod} g(r,s)$, and $h_1$ satisfies some condition of primitivity. These conditions are necessary if $X$ is general with $\mathop {\mathrm{rk}}N(X)=2$. The existence of such kind of a riterion is surprising, and it also gives some geometric interpretation of elements in $N(X)$ with negative square. We describe all irreducible 18-dimensional components of the moduli space of pairs $(X,H)$ with $Y\cong X$. We prove that their number is always infinite. Earlier, similar results have been known only for $r=s$.

Full text: PDF file (296 kB)
References: PDF file   HTML file

English version:
Proceedings of the Steklov Institute of Mathematics, 2004, 246, 204–226

Bibliographic databases:
UDC: 512.7

Citation: V. V. Nikulin, “On Correspondences of a K3 Surface with Itself. I”, Algebraic geometry: Methods, relations, and applications, Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences, Tr. Mat. Inst. Steklova, 246, Nauka, MAIK «Nauka/Inteperiodika», M., 2004, 217–239; Proc. Steklov Inst. Math., 246 (2004), 204–226

Citation in format AMSBIB
\Bibitem{Nik04} \by V.~V.~Nikulin \paper On Correspondences of a~K3 Surface with Itself.~I \inbook Algebraic geometry: Methods, relations, and applications \bookinfo Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences \serial Tr. Mat. Inst. Steklova \yr 2004 \vol 246 \pages 217--239 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm157} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2101295} \zmath{https://zbmath.org/?q=an:1130.14030} \transl \jour Proc. Steklov Inst. Math. \yr 2004 \vol 246 \pages 204--226 

• http://mi.mathnet.ru/eng/tm157
• http://mi.mathnet.ru/eng/tm/v246/p217

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Nikulin V.V., “On correspondences of a K3 surface with itself. II”, Algebraic Geometry, Contemporary Mathematics Series, 422, 2007, 121–172
2. C. G. Madonna, V. V. Nikulin, “Explicit correspondences of a K3 surface with itself”, Izv. Math., 72:3 (2008), 497–508
3. Viacheslav V. Nikulin, “Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups”, Proc. Steklov Inst. Math., 273 (2011), 229–237
4. Madonna C.G., “On Some Moduli Spaces of Bundles on K3 Surfaces, II”, Proc. Amer. Math. Soc., 140:10 (2012), 3397–3408
•  Number of views: This page: 215 Full text: 60 References: 49