higher geometry / derived geometry
geometric little (∞,1)-toposes
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derived smooth geometry
Complex algebraic K3-surfaces admit the structure of elliptic fibrations over the Riemann sphere $\mathbb{C}P^1$. Equipped with such, they are also called elliptic K3-surfaces.
Counted with multiplicity, these elliptic fibrations have 24 singular points (where the elliptic curve-fiber degenerates, to a nodal curve or cuspidal curve).
(review in Huybrechts 16, Section 2-2.4, Propp 18, p. 4).
This number 24 of singularities-with-multiplicities is identified with the Euler number of the topological space underlying the (complex analytic) K3-surface (e.g. Shimada 00, p. 432 (10 of 24), Schütt-Shioda 09, Section 6.7, Theorem 6.10, Huybrechts 16, Chapter 11, Remark 1.12, review includes Marquart 02, Section A.3.3, p. 59).
Up to isomorphism, there are a finite number of possible such elliptic fibrations.
Elliptically fibered Calabi-Yau manifolds play a central role in F-theory:
In passing from M-theory to type IIA string theory, the locus of any Kaluza-Klein monopole in 11d becomes the locus of D6-branes in 10d. The locus of the Kaluza-Klein monopole in turn (as discussed there) is the locus where the $S^1_A$-circle fibration degenerates. Hence in F-theory this is the locus where the fiber of the $S^1_A \times S^1_B$-elliptic fibration degenerates to the nodal curve. Since the T-dual of D6-branes are D7-branes, it follows that D7-branes in F-theory “are” the singular locus of the elliptic fibration.
Now an elliptically fibered complex K3-surface
may be parameterized via the Weierstrass elliptic function as the solution locus of the equation
for $x,y,z \in \mathbb{C}\mathbb{P}^1$, with $f$ a polynomial of degree 8 and $g$ of degree twelve. The j-invariant of the complex elliptic curve which this parameterizes for given $z$ is
The poles $j\to \infty$ of the j-invariant correspond to the nodal curve, and hence it is at these poles that the D7-branes are located.
Since the order of the poles is 24 (the polynomial degree of the discriminant $\Delta = 27 g^2 + 4 f^3$) there are necessarily 24 D7-branes (Sen 96, page 5, Sen 97b, see also Morrison 04, sections 8 and 17, Denef 08, around (3.41), Douglas-Park-Schnell 14).
Under T-duality this translates to 24 D6-branes in type IIA string theory on K3 (Vafa 96, Footnote 2 on p. 6).
Notice that the net charge of these 24 D7-branes is supposed to vanish, due to S-duality effects (e.g. Denef 08, below (3.41)).
Basics:
Robert Friedman, John Morgan, Smooth Four-Manifolds and Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics (1994) (doi:10.1007/978-3-662-03028-8)
Ichiro Shimada, On elliptic K3 surfaces, Michigan Math. J. Volume 47, Issue 3 (2000), 423-446 (euclid:mmj/1030132587)
Matthias Schütt, Tetsuji Shioda, Section 12 of: Elliptic surfaces, Adv. Stud. Pure Math. Algebraic Geometry in East Asia — Seoul 2008, J. H. Keum, S. Kondō, K. Konno and K. Oguiso, eds. (Tokyo: Mathematical Society of Japan, 2010), 51 - 160 (arXiv:0907.0298, euclid:aspm/1543085637)
Daniel Huybrechts, Chapter 11 of: Lectures on K3-surfaces, Cambridge University Press 2016 (pdf, pdf, doi:10.1017/CBO9781316594193)
Further review:
Tristan Hübsch, Section 6.2 of: Calabi-Yau Manifolds – A Bestiary for Physicists, World Scientific 1992 (doi:10.1142/1410)
Monika Marquart, Section A.3 of: Applications of Dualities in String Theory, 2002 (sundoc:02/02H150, pdf)
Classification:
Viacheslav Nikulin, Elliptic fibrations on K3 surfaces, Proceedings of the Edinburgh Mathematical Society, Volume 57 Issue 1 (arXiv:1010.3904, doi:10.1017/S0013091513000953)
O. Lecacheux, Weierstrass Equations for the Elliptic Fibrations of a K3 Surface In: Balakrishnan J., Folsom A., Lalín M., Manes M. (eds.) Research Directions in Number Theory Association for Women in Mathematics Series, vol 19. Springer (2019) (doi:10.1007/978-3-030-19478-9_4)
Marie Bertin, Elliptic Fibrations on K3 surfaces, 2013 (pdf)
Discussion in the context of K3-spectra: