Cubic 4-folds

Number of rows: (20–1000)

$d$	$κ$	degree (twisted) K3
8	$- \infty$	$(2, 2)$
12	$- \infty$
14	$- \infty$	$14$
18	$- \infty$	$(2, 3)$
20	$- \infty$
24	$- \infty$	$(6, 2)$
26	$- \infty$	$26$
30	$- \infty$
32	$- \infty$	$(2, 4)$
36	$- \infty$
38	$- \infty$	$38$
42	$- \infty$	$42$
44	$- \infty$
48
50		$(2, 5)$
54		$(6, 3)$
56		$(14, 2)$
60
62		$62$
66
68
72		$(2, 6)$
74		$74$
78		$78$
80
84
86	$\geq 0$	$86$
90
92
96		$(6, 4)$
98	$\geq 0$	$98$ , $(2, 7)$
102	$\geq 0$
104	$\geq 0$	$(26, 2)$
108
110	$\geq 0$
114	$19$	$114$
116	$19$
120
122	$\geq 0$	$122$
126	$19$	$(14, 3)$
128	$\geq 0$	$(2, 8)$
132
134	$19$	$134$
138	$\geq 0$
140	$19$
144	$19$
146	$19$	$146$
150	$\geq 0$	$(6, 5)$
152	$\geq 0$	$(38, 2)$
156	$19$
158	$19$	$158$
162	$19$	$(2, 9)$
164	$19$
168	$19$	$(42, 2)$
170	$19$
174	$19$
176	$19$
180
182	$19$	$182$
186	$19$	$186$
188	$19$
192	$\geq 0$
194	$19$	$194$
198	$19$
200	$19$	$(2, 10)$
204	$19$
206	$19$	$206$
210	$19$
212	$19$
216	$19$	$(6, 6)$
218	$19$	$218$
222	$19$	$222$
224	$19$	$(14, 4)$
228	$19$
230	$19$
234	$19$	$(26, 3)$
236	$19$
240	$19$
242	$19$	$(2, 11)$
246	$19$
248	$19$	$(62, 2)$
252	$19$
254	$19$	$254$
258	$19$	$258$
260	$19$
264	$19$
266	$19$	$266$
270	$19$
272	$19$
276	$19$
278	$19$	$278$
282	$19$
284	$19$
288	$19$	$(2, 12)$
290	$19$
294	$19$	$294$ , $(6, 7)$
296	$19$	$(74, 2)$
300	$19$
302	$19$	$302$
306	$19$
308	$19$
312	$19$	$(78, 2)$
314	$19$	$314$
318	$19$
320	$19$
324	$19$
326	$19$	$326$
330	$19$
332	$19$
336	$19$
338	$19$	$338$ , $(2, 13)$
342	$19$	$(38, 3)$
344	$19$	$(86, 2)$
348	$19$
350	$19$	$(14, 5)$
354	$19$
356	$19$
360	$19$
362	$19$	$362$
366	$19$	$366$
368	$19$
372	$19$
374	$19$
378	$19$	$(42, 3)$
380	$19$
384	$19$	$(6, 8)$
386	$19$	$386$
390	$19$
392	$19$	$(98, 2)$ , $(2, 14)$
396	$19$
398	$19$	$398$
402	$19$	$402$
404	$19$
408	$19$
410	$19$
414	$19$
416	$19$	$(26, 4)$
420	$19$
422	$19$	$422$
426	$19$
428	$19$
432	$19$
434	$19$	$434$
438	$19$	$438$
440	$19$
444	$19$
446	$19$	$446$
450	$19$	$(2, 15)$
452	$19$
456	$19$	$(114, 2)$
458	$19$	$458$
462	$19$
464	$19$
468	$19$
470	$19$
474	$19$	$474$
476	$19$
480	$19$
482	$19$	$482$
486	$19$	$(6, 9)$
488	$19$	$(122, 2)$
492	$19$
494	$19$	$494$
498	$19$
500	$19$
504	$19$	$(14, 6)$
506	$19$
510	$19$
512	$19$	$(2, 16)$
516	$19$
518	$19$	$518$
522	$19$
524	$19$
528	$19$
530	$19$
534	$19$
536	$19$	$(134, 2)$
540	$19$
542	$19$	$542$
546	$19$	$546$
548	$19$
552	$19$
554	$19$	$554$
558	$19$	$(62, 3)$
560	$19$
564	$19$
566	$19$	$566$
570	$19$
572	$19$
576	$19$
578	$19$	$(2, 17)$
582	$19$	$582$
584	$19$	$(146, 2)$
588	$19$
590	$19$
594	$19$
596	$19$
600	$19$	$(6, 10)$
602	$19$	$602$
606	$19$
608	$19$	$(38, 4)$
612	$19$
614	$19$	$614$
618	$19$	$618$
620	$19$
624	$19$
626	$19$	$626$
630	$19$
632	$19$	$(158, 2)$
636	$19$
638	$19$
642	$19$
644	$19$
648	$19$	$(2, 18)$
650	$19$	$(26, 5)$
654	$19$	$654$
656	$19$
660	$19$
662	$19$	$662$
666	$19$	$(74, 3)$
668	$19$
672	$19$	$(42, 4)$
674	$19$	$674$
678	$19$
680	$19$
684	$19$
686	$19$	$686$ , $(14, 7)$
690	$19$
692	$19$
696	$19$
698	$19$	$698$
702	$19$	$(78, 3)$
704	$19$
708	$19$
710	$19$
714	$19$
716	$19$
720	$19$
722	$19$	$722$ , $(2, 19)$
726	$19$	$(6, 11)$
728	$19$	$(182, 2)$
732	$19$
734	$19$	$734$
738	$19$
740	$19$
744	$19$	$(186, 2)$
746	$19$	$746$
750	$19$
752	$19$
756	$19$
758	$19$	$758$
762	$19$	$762$
764	$19$
768	$19$
770	$19$
774	$19$	$(86, 3)$
776	$19$	$(194, 2)$
780	$19$
782	$19$
786	$19$
788	$19$
792	$19$
794	$19$	$794$
798	$19$	$798$
800	$19$	$(2, 20)$
804	$19$
806	$19$	$806$
810	$19$
812	$19$
816	$19$
818	$19$	$818$
822	$19$
824	$19$	$(206, 2)$
828	$19$
830	$19$
834	$19$	$834$
836	$19$
840	$19$
842	$19$	$842$
846	$19$
848	$19$
852	$19$
854	$19$	$854$
858	$19$
860	$19$
864	$19$	$(6, 12)$
866	$19$	$866$
870	$19$
872	$19$	$(218, 2)$
876	$19$
878	$19$	$878$
882	$19$	$(98, 3)$ , $(2, 21)$
884	$19$
888	$19$	$(222, 2)$
890	$19$
894	$19$
896	$19$	$(14, 8)$
900	$19$
902	$19$
906	$19$	$906$
908	$19$
912	$19$
914	$19$	$914$
918	$19$
920	$19$
924	$19$
926	$19$	$926$
930	$19$
932	$19$
936	$19$	$(26, 6)$
938	$19$	$938$
942	$19$	$942$
944	$19$
948	$19$
950	$19$	$(38, 5)$
954	$19$
956	$19$
960	$19$
962	$19$	$962$
966	$19$
968	$19$	$(2, 22)$
972	$19$
974	$19$	$974$
978	$19$	$978$
980	$19$
984	$19$
986	$19$
990	$19$
992	$19$	$(62, 4)$
996	$19$
998	$19$	$998$

References

Addington, N. (2016). On two rationality conjectures for cubic fourfolds. Math. Res. Lett., 23(1), 1–13. https://doi.org/10.4310/MRL.2016.v23.n1.a1

Farkas, G., & Verra, A. (2018). The universal K3 surface of genus 14 via cubic fourfolds. J. Math. Pures Appl. (9), 111, 1–20. https://doi.org/10.1016/j.matpur.2017.07.014

Hassett, B. (2000). Special cubic fourfolds. Compositio Math., 120(1), 1–23. https://doi.org/10.1023/A:1001706324425

Huybrechts, D. (2017). The K3 category of a cubic fourfold. Compos. Math., 153(3), 586–620. https://doi.org/10.1112/S0010437X16008137

Huybrechts, D. (2022). The K3 category of a cubic fourfold – an update. http://www.math.uni-bonn.de/people/huybrech/CubicPaperAnUpdate.pdf

Tanimoto, S., & Várilly-Alvarado, A. (2019). Kodaira dimension of moduli of special cubic fourfolds. J. Reine Angew. Math., 752, 265–300. https://doi.org/10.1515/crelle-2016-0053

Explanation

$(*)$

Hassett's condition for non-emptiness of $C_{d}$

$d$ satisfied $d \geq 8$ and $d \equiv 0, 2 (\mod 6)$

$(* *)$

Hassett's condition for an associated K3 surface

$d$ satisfies $(*)$ and $d$ is not divisible by $4$ , $9$ or any odd prime $p \equiv 2 (\mod 3)$

there exists a K3 surface $S$ such that $\tilde{H} (X, Z) ≅ \tilde{H} (S, Z)$

there exists a K3 surface $S$ such that $S$ is birational to a moduli space of stable sheaves on $S$

$(* *^{'})$

Huybrechts' condition for an associated K3 twisted surface

$d$ satisfies $(*)$ and in the prime factorisation $d / 2 = \prod p_{i}^{n_{i}}$ one has $n_{i} \equiv 0 (\mod 2)$ for all primes $p_{i} \equiv 2 (\mod 3)$

there exists a K3 surface $S$ and a Brauer class $α \in Br (S)$ such that $\tilde{H} (X, Z) ≅ \tilde{H} (S, α, Z)$

$(* * *)$

Addington's condition for an associated K3 surface

$d$ satisfies $(*)$ and $d$ is of the form $\frac{2 n^{2} + 2 n + 2}{a^{2}}$ for some $n, a \in Z$

there exists a K3 surface $S$ such that $F (X)$ is birational to ${Hilb}^{2} S$

We have that $(* * *) \Rightarrow (* *) \Rightarrow (* *^{'})$

About

This is an overview of (some of) what we know about the relationship between cubic fourfolds and K3 surfaces.

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