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 Tr. Mat. Inst. Steklova, 2003, Volume 241, Pages 132–168 (Mi tm393)

On a Classical Correspondence between K3 Surfaces

C. G. Madonnaa, V. V. Nikulinbc

a Università degli Studi di Roma — Tor Vergata
b Steklov Mathematical Institute, Russian Academy of Sciences
c University of Liverpool

Abstract: Let $X$ be a K3 surface that is the intersection (i.e. a net $\mathbb P^2$) of three quadrics in $\mathbb P^5$. The curve of degenerate quadrics has degree 6 and defines a natural double covering $Y$ of $\mathbb P^2$ ramified in this curve which is again a K3. This is a classical example of a correspondence between K3 surfaces that is related to the moduli of sheaves on K3 studied by Mukai. When are general (for fixed Picard lattices) $X$ and $Y$ isomorphic? We give necessary and sufficient conditions in terms of Picard lattices of $X$ and $Y$. For example, for the Picard number 2, the Picard lattice of $X$ and $Y$ is defined by its determinant $-d$, where $d>0$, $d\equiv 1\mod 8$, and one of the equations $a^2-db^2=8$ or $a^2-db^2=-8$ has an integral solution $(a,b)$. Clearly, the set of these $d$ is infinite: $d\in \{(a^2\mp 8)/b^2\}$, where $a$ and $b$ are odd integers. This gives all possible divisorial conditions on the 19-dimensional moduli of intersections of three quadrics $X$ in $\mathbb P^5$, which imply $Y\cong X$. One of them, when $X$ has a line, is classical and corresponds to $d=17$. Similar considerations can be applied to a realization of an isomorphism $(T(X)\otimes \mathbb Q, H^{2,0}(X)) \cong (T(Y)\otimes \mathbb Q, H^{2,0}(Y))$ of transcendental periods over $\mathbb Q$ of two K3 surfaces $X$ and $Y$ by a fixed sequence of types of Mukai vectors.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 241, 120–153

Bibliographic databases:
UDC: 512.7

Citation: C. G. Madonna, V. V. Nikulin, “On a Classical Correspondence between K3 Surfaces”, Number theory, algebra, and algebraic geometry, Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich, Tr. Mat. Inst. Steklova, 241, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 132–168; Proc. Steklov Inst. Math., 241 (2003), 120–153

Citation in format AMSBIB
\Bibitem{MadNik03} \by C.~G.~Madonna, V.~V.~Nikulin \paper On a~Classical Correspondence between K3 Surfaces \inbook Number theory, algebra, and algebraic geometry \bookinfo Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich \serial Tr. Mat. Inst. Steklova \yr 2003 \vol 241 \pages 132--168 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm393} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2024049} \zmath{https://zbmath.org/?q=an:1076.14046} \transl \jour Proc. Steklov Inst. Math. \yr 2003 \vol 241 \pages 120--153 

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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. V. V. Nikulin, “On Correspondences of a K3 Surface with Itself. I”, Proc. Steklov Inst. Math., 246 (2004), 204–226
2. Madonna C.G., “On some moduli spaces of bundles on $K3$ surfaces”, Monatsh. Math., 146:4 (2005), 333–339
3. Dujella A., Franušić Z., “On differences of two squares in some quadratic fields”, Rocky Mountain J. Math., 37:2 (2007), 429–453
4. Cynk S., Rams S., “On a map between two K3 surfaces associated to a net of quadrics”, Arch. Math. (Basel), 88:2 (2007), 109–122
5. Nikulin V.V., “On correspondences of a K3 surface with itself. II”, Algebraic Geometry, Contemporary Mathematics Series, 422, 2007, 121–172
6. C. G. Madonna, V. V. Nikulin, “Explicit correspondences of a K3 surface with itself”, Izv. Math., 72:3 (2008), 497–508
7. 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
8. Michalek M., “Birational Maps Between Calabi-Yau Manifolds Associated to Webs of Quadrics”, J. Algebra, 370 (2012), 186–197
9. Madonna C.G., “On Some Moduli Spaces of Bundles on K3 Surfaces, II”, Proc. Amer. Math. Soc., 140:10 (2012), 3397–3408
10. Cynk S., Rams S., “on Calabi-Yau Threefolds Associated To a Web of Quadrics”, Forum Math., 27:2 (2015), 699–734
11. Kuznetsov A., Shinder E., “Grothendieck Ring of Varieties, D- and l-Equivalence, and Families of Quadrics”, Sel. Math.-New Ser., 24:4 (2018), 3475–3500
12. V. A. Krasnov, “On a classical correspondence of real K3 surfaces”, Izv. Math., 82:4 (2018), 662–693
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