Gorenstein Formats, Canonical and Calabi–Yau Threefolds

Figures & data

Table 3. Codimension three.

Variety	Basket $\mathcal{B}$	$K_X^3$	χ	$K_X{c}_2$	w	Syz weights
$\begin{array}{c}{X}_{3^4,4}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^7)\hfill \end{array}$		20	–6	144	$(0,1,1,1,1)$	$\begin{array}{cccc}1& 1& 1& 1\\ & 2& 2& 2\\ & & 2& 2\\ & & & 2\end{array}$
$\begin{array}{c}{X}_{3^2,4^3}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^6,2)\hfill \end{array}$		14	–5	120	$\frac{1}{2}(1,1,1,3,3)$	$\begin{array}{cccc}1& 1& 2& 2\\ & 1& 2& 2\\ & & 2& 2\\ & & & 3\end{array}$
$\begin{array}{c}{X}_{3,4^3,5}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^5,2^2)\hfill \end{array}$	$\frac{1}{2}(1,1,1)$	$\frac{19}{2}$	–4	$\frac{195}{2}$	$(0,1,1,1,2)$	$\begin{array}{cccc}1& 1& 1& 2\\ & 2& 2& 3\\ & & 2& 3\\ & & & 3\end{array}$
$\begin{array}{c}{X}_{4^5}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^5,2^2)\hfill \end{array}$		10	–4	96	$(1,1,1,1,1)$	$\begin{array}{cccc}2& 2& 2& 2\\ & 2& 2& 2\\ & & 2& 2\\ & & & 2\end{array}$
$\begin{array}{c}{X}_{4^3,5^2}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^4,2^3)\hfill \end{array}$	$3\times \frac{1}{2}(1,1,1)$	$\frac{13}{2}$	–3	$\frac{153}{2}$	$\frac{1}{2}(1,1,3,3,3)$	$\begin{array}{cccc}1& 2& 2& 2\\ & 2& 2& 2\\ & & 3& 3\\ & & & 3\end{array}$
$\begin{array}{c}{X}_{4^2,5^2,6}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^4,2^2,3)\hfill \end{array}$	$\frac{1}{2}(1,1,1)$	$\frac{11}{2}$	–3	$\frac{147}{2}$	$(0,1,1,2,2)$	$\begin{array}{cccc}1& 1& 2& 2\\ & 2& 3& 3\\ & & 3& 3\\ & & & 4\end{array}$
$\begin{array}{c}{X}_{4,5^2,6^2}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^3,2^3,3)\hfill \end{array}$	$5\times \frac{1}{2}(1,1,1)$	$\frac{7}{2}$	–2	$\frac{111}{2}$	$\frac{1}{2}(1,1,3,3,5)$	$\begin{array}{cccc}1& 2& 2& 3\\ & 2& 2& 3\\ & & 3& 4\\ & & & 4\end{array}$
$\begin{array}{c}{X}_{4,5,6^2,7}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^3,2^2,3^2)\hfill \end{array}$	$\frac{1}{2}(1,1,1),\frac{1}{3}(1,2,2)$	$\frac{17}{6}$	–2	$\frac{313}{6}$	$(0,1,1,2,3)$	$\begin{array}{cccc}1& 1& 2& 3\\ & 2& 3& 4\\ & & 3& 4\\ & & & 5\end{array}$
$\begin{array}{c}{X}_{5^2,6^3}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^3,2^2,3^2)\hfill \end{array}$	$2\times \frac{1}{2}(1,1,1)$	3	–2	51	$(1,1,1,2,2)$	$\begin{array}{cccc}2& 2& 3& 3\\ & 2& 3& 3\\ & & 3& 3\\ & & & 4\end{array}$
$\begin{array}{c}{X}_{5,6^3,7}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^2,2^3,3^2)\hfill \end{array}$	$7\times \frac{1}{2}(1,1,1),\frac{1}{3}(1,2,2)$	$\frac{11}{6}$	–1	$\frac{223}{6}$	$\frac{1}{2}(1,3,3,3,5)$	$\begin{array}{cccc}2& 2& 2& 3\\ & 3& 3& 4\\ & & 3& 4\\ & & & 4\end{array}$
$\begin{array}{c}{X}_{6^3,7^2}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^2,2^2,3^3)\hfill \end{array}$	$\frac{1}{2}(1,1,1),3\times \frac{1}{3}(1,2,2)$	$\frac{3}{2}$	–1	$\frac{67}{2}$	$(1,1,2,2,2)$	$\begin{array}{cccc}2& 3& 3& 3\\ & 3& 3& 3\\ & & 4& 4\\ & & & 4\end{array}$
$\begin{array}{c}{X}_{6^2,7^2,8}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^2,2^2,3^2,4)\hfill \end{array}$	$4\times \frac{1}{2}(1,1,1),\frac{1}{3}(1,2,2)$	$\frac{4}{3}$	–1	$\frac{98}{3}$	$\frac{1}{2}(1,3,3,5,5)$	$\begin{array}{cccc}2& 2& 3& 3\\ & 3& 4& 4\\ & & 4& 4\\ & & & 5\end{array}$
$\begin{array}{c}{X}_{6,7,8,9,10}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^2,2,3^2,4,5)\hfill \end{array}$	$\frac{1}{2}(1,1,1),\frac{1}{3}(1,2,2)$	$\frac{5}{6}$	–1	$\frac{169}{6}$	$(0,1,2,3,4)$	$\begin{array}{cccc}1& 2& 3& 4\\ & 3& 4& 5\\ & & 5& 6\\ & & & 7\end{array}$
$\begin{array}{c}{X}_{7,8^2,9,10}\hfill \\ \hspace{1em}\subset \mathbb{P}(1,2^2,3^2,4,5)\hfill \end{array}$	$7\times \frac{1}{2}(1,1,1),3\times \frac{1}{3}(1,2,2)$	$\frac{1}{2}$	0	$\frac{37}{2}$	$\frac{1}{2}(1,3,5,5,7)$	$\begin{array}{cccc}2& 3& 3& 4\\ & 4& 4& 5\\ & & 5& 6\\ & & & 6\end{array}$
$\begin{array}{c}{X}_{8,9^2,{10}^2}\hfill \\ \hspace{1em}\subset \mathbb{P}(1,2,3^2,4^2,5)\hfill \end{array}$	$3\times \frac{1}{2}(1,1,1),\frac{1}{3}(1,2,2),2\times \frac{1}{4}(1,3,3)$	$\frac{1}{3}$	0	$\frac{44}{3}$	$\frac{1}{2}(3,3,5,5,7)$	$\begin{array}{cccc}3& 4& 4& 5\\ & 4& 4& 5\\ & & 5& 6\\ & & & 6\end{array}$
$\begin{array}{c}{X}_{8,9,{10}^2,11}\hfill \\ \hspace{1em}\subset \mathbb{P}(1,2,3^2,4,5^2)\hfill \end{array}$	$\frac{1}{2}(1,1,1),3\times \frac{1}{3}(1,2,2),\frac{1}{5}(2,3,4)$	$\frac{3}{10}$	0	$\frac{143}{10}$	$(1,2,2,3,4)$	$\begin{array}{cccc}3& 3& 4& 5\\ & 4& 5& 6\\ & & 5& 6\\ & & & 7\end{array}$
$\begin{array}{c}{X}_{12,13,14,15,16}\hfill \\ \hspace{1em}\subset \mathbb{P}(1,3,4,5,6,7,8)\hfill \end{array}$	$\frac{1}{2}(1,1,1),\frac{1}{3}(1,2,2),\frac{1}{4}(1,3,3)$	$\frac{1}{12}$	0	$\frac{95}{12}$	$\frac{1}{2}(3,5,7,9,11)$	$\begin{array}{cccc}4& 5& 6& 7\\ & 6& 7& 8\\ & & 8& 9\\ & & & 10\end{array}$
$\begin{array}{c}{X}_{12,13,14,15,16}\hfill \\ \hspace{1em}\subset \mathbb{P}(3,4^2,5^2,6,7)\hfill \end{array}$	$2\times \frac{1}{2}(1,1,1),\frac{1}{3}(1,2,2),2\times \frac{1}{4}(1,3,3),2\times \frac{1}{5}(1,4,4),\frac{1}{5}(2,3,4)$	$\frac{1}{30}$	1	$\frac{107}{30}$	$\frac{1}{2}(3,5,7,9,11)$	$\begin{array}{cccc}4& 5& 6& 7\\ & 6& 7& 8\\ & & 8& 9\\ & & & 10\end{array}$

Table 1. The number of cases of Fano, Calabi–Yau, and canonical 3-dimensional orbifolds in various formats. All were computed allowing isolated canonical quotient singularities. The column $k_{\text{last}}$ gives the largest adjunction number for which a result was found; $k_{\text{max}}$ gives the largest degree searched; #raw gives the number of candidates found by the computer; #results gives the number of candidates after removing obvious failures. (The 317 Calabi–Yau hypersurfaces are taken from [Kreuzer and Skarke 00] for completeness, since the method we use here is not effective in that case.)

Dim	k	Codim	Format	Reference	$k_{\text{last}}$	$k_{\text{max}}$	#raw	#results
3	–1	1	c.i.	[Iskovskikh and Prokhorov 99]	66	90	95	95
		2	c.i.	[Iskovskikh and Prokhorov 99]	54	124	85	85
		3	c.i.	Classical	6	77	1	1
		3	Gr(2,5)	[Altınok 05]	45	70	69	69
		4	Gr(2,5) ∩ H	Classical	7	45	1	1
		5	OGr(5,10)	Classical	4	73	1	1
3	0	1	c.i.	[Kreuzer and Skarke 00]				317
		2	c.i.		120	121	419	401
		3	c.i.		74	77	25	22
		3	Gr(2,5)		71	71	226	187
		4	c.i.	Classical	8	32	1	1
		4	Gr(2,5) ∩ H		39	46	123	14
		5	OGr(5,10)		44	46	23	23
3	1	1	c.i.	[Iskovskikh and Prokhorov 99]	46	85	23	23
		2	c.i.	[Iskovskikh and Prokhorov 99]	40	130	66	59
		3	c.i.	[Iskovskikh and Prokhorov 99]	46	80	38	37
		3	Gr(2,5)	Theorem 1.1	35	71	18	18
		4	c.i.	Classical	9	34	1	1
		4	Gr(2,5) ∩ H	Theorem 1.2	41	46	84	57
		5	c.i.	Classical	10	30	1	1
		5	OGr(5,10)	Theorem 1.2	32	74	21	21

Table 2. Examples of surfaces S of general type, polarized by $A=\frac{1}{k}{K}_S,$ in various formats: #results gives the number of numerical types that arise early in the search, and the right-most column lists these as pairs of invariants, p_g and $K_S^2,$ that are realized by surfaces. The general member of each family with k = 1 is smooth; $\mathbb{Z}/2$ canonical quotient points (A₁ singularities where A is not Cartier) often appear when k = 2.

dim	k	Codim	Format	#Results	Pairs of invariants $(p_g,K_S^2)$ that are realized
2	1	1	c.i.	4	$(p_g,K_S^2)=$ (2,1); (3,2); (3,3); (4,5)
		2	c.i.	6	(1,1); (2,2); (3,4); (4,4); (5,8); (5,9)
		3	c.i.	1	(6,12)
		3	Gr(2,5)	5	(3,5); (4,7); (5,10); (6,11); (6,13)
		4	c.i.	1	(7,16)
		4	Gr(2,5) ∩ H	3	(6,10); (7,15); (7,16)
2	2	1	c.i.	8	(2,2); (3,2); (4,4); (4,6); (5,8); (6,8); (7,14); (10,24)
		2	c.i.	17	(3,2); (3,4); (4,4); (4,6); (4,8); (5,6); (5,10); …
		3	c.i.	9	(4,4); (5,8); (7,16); (9,24); …
		3	Gr(2,5)	18	(5,12); (6,14); (6,10); (7,16); (8,22); …
		4	c.i.	1	(25,96)
		4	Gr(2,5) ∩ H	17	(6,10); (7,14); (9, 18); (10,20); …
		5	c.i.	1	(31,128)
		5	OGr(5,10)	3	(9,24); (14,46); (22,84)

Table 4. Codimension five.

Variety	Basket $\mathcal{B}$	$K_X^3$	χ	$K_X{c}_2$	u and w	Variable weights $x,x_i,x_{\mathit{ij}}$
$\begin{array}{c}{X}_{2,3^8,4}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^7,2^2)\hfill \end{array}$	$2\times \frac{1}{2}(1,1,1)$	21	–6	147	$\begin{array}{c}1\\ (0,0,0,0,1)\end{array}$	$\begin{array}{c}1\\ 2,2,2,2,1\end{array}$	$\begin{array}{cccc}1& 1& 1& 2\\ & 1& 1& 2\\ & & 1& 2\\ & & & 2\end{array}$
$\begin{array}{c}{X}_{3^5,4^5}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^5,2^4)\hfill \end{array}$	$5\times \frac{1}{2}(1,1,1)$	$\frac{23}{2}$	–4	$\frac{207}{2}$	$\begin{array}{c}1\\ \frac{1}{2}(-1,1,1,1,1)\end{array}$	$\begin{array}{c}1\\ 3,2,2,2,2\end{array}$	$\begin{array}{cccc}1& 1& 1& 1\\ & 2& 2& 2\\ & & 2& 2\\ & & & 2\end{array}$
$\begin{array}{c}{X}_{3^5,4^5}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^6,2^2,3)\hfill \end{array}$	$\frac{1}{3}(1,2,2)$	$\frac{46}{3}$	–5	$\frac{368}{3}$	$\begin{array}{c}1\\ \frac{1}{2}(-1,1,1,1,1)\end{array}$	$\begin{array}{c}1\\ 3,2,2,2,2\end{array}$	$\begin{array}{cccc}1& 1& 1& 1\\ & 2& 2& 2\\ & & 2& 2\\ & & & 2\end{array}$
$\begin{array}{c}{X}_{3^2,4^6,5^2}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^4,2^4,3)\hfill \end{array}$	$4\times \frac{1}{2}(1,1,1),\frac{1}{3}(1,2,2)$	$\frac{22}{3}$	–3	$\frac{242}{3}$	$\begin{array}{c}1\\ (0,0,0,1,1)\end{array}$	$\begin{array}{c}1\\ 3,3,3,2,2\end{array}$	$\begin{array}{cccc}1& 1& 2& 2\\ & 1& 2& 2\\ & & 2& 2\\ & & & 3\end{array}$
$\begin{array}{c}{X}_{4^{10}}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^3,2^6)\hfill \end{array}$	$12\times \frac{1}{2}(1,1,1)$	6	–2	66	$\begin{array}{c}2\\ (0,0,0,0,0)\end{array}$	$\begin{array}{c}2\\ 2,2,2,2,2\end{array}$	$\begin{array}{cccc}2& 2& 2& 2\\ & 2& 2& 2\\ & & 2& 2\\ & & & 2\end{array}$
$\begin{array}{c}{X}_{4^3,5^4,6^3}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^3,2^3,3^2,4)\hfill \end{array}$	$3\times \frac{1}{2}(1,1,1),\frac{1}{4}(1,3,3)$	$\frac{15}{4}$	–2	$\frac{225}{4}$	$\begin{array}{c}1\\ (0,0,1,1,1)\end{array}$	$\begin{array}{c}1\\ 4,4,3,3,3\end{array}$	$\begin{array}{cccc}1& 2& 2& 2\\ & 2& 2& 2\\ & & 3& 3\\ & & & 3\end{array}$
$\begin{array}{c}{X}_{4^3,5^4,6^3}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^2,2^4,3^3)\hfill \end{array}$	$7\times \frac{1}{2}(1,1,1),3\times \frac{1}{3}(1,2,2)$	$\frac{5}{2}$	–1	$\frac{85}{2}$	$\begin{array}{c}1\\ (0,0,1,1,1)\end{array}$	$\begin{array}{c}1\\ 4,4,3,3,3\end{array}$	$\begin{array}{cccc}1& 2& 2& 2\\ & 2& 2& 2\\ & & 3& 3\\ & & & 3\end{array}$
$\begin{array}{c}{X}_{4,5^2,6^4,7^2,8}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^2,2^3,3^2,4,5)\hfill \end{array}$	$6\times \frac{1}{2}(1,1,1),\frac{1}{5}(2,3,4)$	$\frac{9}{5}$	–1	$\frac{189}{5}$	$\begin{array}{c}1\\ (0,0,1,1,2)\end{array}$	$\begin{array}{c}1\\ 5,5,4,4,3\end{array}$	$\begin{array}{cccc}1& 2& 2& 3\\ & 2& 2& 3\\ & & 3& 4\\ & & & 4\end{array}$
$\begin{array}{c}{X}_{4,5^2,6^4,7^2,8}\hfill \\ \hspace{1em}\subset \mathbb{P}(1^2,2^2,3^3,4^2)\hfill \end{array}$	$2\times \frac{1}{2}(1,1,1),2\times \frac{1}{4}(1,3,3)$	$\frac{3}{2}$	–1	$\frac{69}{2}$	$\begin{array}{c}1\\ (0,0,1,1,2)\end{array}$	$\begin{array}{c}1\\ 5,5,4,4,3\end{array}$	$\begin{array}{cccc}1& 2& 2& 3\\ & 2& 2& 3\\ & & 3& 4\\ & & & 4\end{array}$
$\begin{array}{c}{X}_{4,5^2,6^4,7^2,8}\hfill \\ \hspace{1em}\subset \mathbb{P}(1,2^3,3^4,4)\hfill \end{array}$	$6\times \frac{1}{2}(1,1,1),6\times \frac{1}{3}(1,2,2)$	1	0	25	$\begin{array}{c}1\\ (0,0,1,1,2)\end{array}$	$\begin{array}{c}1\\ 5,5,4,4,3\end{array}$	$\begin{array}{cccc}1& 2& 2& 3\\ & 2& 2& 3\\ & & 3& 4\\ & & & 4\end{array}$
$\begin{array}{c}{X}_{5^2,6^6,7^2}\hfill \\ \hspace{1em}\subset \mathbb{P}(1,2^3,3^4,4)\hfill \end{array}$	$7\times \frac{1}{2}(1,1,1),4\times \frac{1}{3}(1,2,2),\frac{1}{4}(1,3,3)$	$\frac{13}{12}$	0	$\frac{299}{12}$	$\begin{array}{c}2\\ (0,0,0,1,1)\end{array}$	$\begin{array}{c}2\\ 4,4,4,3,3\end{array}$	$\begin{array}{cccc}2& 2& 3& 3\\ & 2& 3& 3\\ & & 3& 3\\ & & & 4\end{array}$
$\begin{array}{c}{X}_{6^3,7^4,8^3}\hfill \\ \hspace{1em}\subset \mathbb{P}(1,2^2,3^3,4^2,5)\hfill \end{array}$	$5\times \frac{1}{2}(1,1,1),3\times \frac{1}{3}(1,2,2),\frac{1}{5}(1,4,4)$	$\frac{7}{10}$	0	$\frac{203}{10}$	$\begin{array}{c}2\\ (0,0,1,1,1)\end{array}$	$\begin{array}{c}2\\ 5,5,4,4,4\end{array}$	$\begin{array}{cccc}2& 3& 3& 3\\ & 3& 3& 3\\ & & 4& 4\\ & & & 4\end{array}$
$\begin{array}{c}{X}_{6,7^2,8^4,9^2,10}\hfill \\ \hspace{1em}\subset \mathbb{P}(2^2,3^3,4^2,5^2)\hfill \end{array}$	$8\times \frac{1}{2}(1,1,1),5\times \frac{1}{3}(1,2,2),2\times \frac{1}{5}(2,3,4)$	$\frac{4}{15}$	1	$\frac{164}{15}$	$\begin{array}{c}2\\ (0,0,1,1,2)\end{array}$	$\begin{array}{c}2\\ 6,6,5,5,4\end{array}$	$\begin{array}{cccc}2& 3& 3& 4\\ & 3& 3& 4\\ & & 4& 5\\ & & & 5\end{array}$
$\begin{array}{c}{X}_{6,7^2,8^4,9^2,10}\hfill \\ \hspace{1em}\subset \mathbb{P}(1,2,3^2,4^3,5^2)\hfill \end{array}$	$4\times \frac{1}{2}(1,1,1),2\times \frac{1}{5}(1,4,4)$	$\frac{2}{5}$	0	$\frac{78}{5}$	$\begin{array}{c}2\\ (0,0,1,1,2)\end{array}$	$\begin{array}{c}2\\ 6,6,5,5,4\end{array}$	$\begin{array}{cccc}2& 3& 3& 4\\ & 3& 3& 4\\ & & 4& 5\\ & & & 5\end{array}$
$\begin{array}{c}{X}_{6,7,8^2,9^2,{10}^2,11,12}\hfill \\ \hspace{1em}\subset \mathbb{P}(1,2,3^2,4^2,5,6,7)\hfill \end{array}$	$3\times \frac{1}{2}(1,1,1),\frac{1}{4}(1,3,3),\frac{1}{7}(3,4,6)$	$\frac{9}{28}$	0	$\frac{423}{28}$	$\begin{array}{c}1\\ (0,1,1,2,3)\end{array}$	$\begin{array}{c}1\\ 8,7,7,6,5\end{array}$	$\begin{array}{cccc}2& 2& 3& 4\\ & 3& 4& 5\\ & & 4& 5\\ & & & 6\end{array}$
$\begin{array}{c}{X}_{8^3,9^4,{10}^3}\hfill \\ \hspace{1em}\subset \mathbb{P}(2,3^2,4^3,5^3)\hfill \end{array}$	$4\times \frac{1}{2}(1,1,1),3\times \frac{1}{4}(1,3,3),3\times \frac{1}{5}(2,3,4)$	$\frac{3}{20}$	1	$\frac{153}{20}$	$\begin{array}{c}3\\ (0,0,1,1,1)\end{array}$	$\begin{array}{c}3\\ 6,6,5,5,5\end{array}$	$\begin{array}{cccc}3& 4& 4& 4\\ & 4& 4& 4\\ & & 5& 5\\ & & & 5\end{array}$
$\begin{array}{c}{X}_{8,9^2,{10}^4,{11}^2,12}\hfill \\ \hspace{1em}\subset \mathbb{P}(2,3^2,4^2,5^2,6,7)\hfill \end{array}$	$4\times \frac{1}{2}(1,1,1),4\times \frac{1}{3}(1,2,2),2\times \frac{1}{4}(1,3,3),\frac{1}{7}(2,5,6)$	$\frac{5}{42}$	1	$\frac{295}{42}$	$\begin{array}{c}3\\ (0,0,1,1,2)\end{array}$	$\begin{array}{c}3\\ 7,7,6,6,5\end{array}$	$\begin{array}{cccc}3& 4& 4& 5\\ & 4& 4& 5\\ & & 5& 6\\ & & & 6\end{array}$
$\begin{array}{c}{X}_{8,9,{10}^2,{11}^2,{12}^2,13,14}\hfill \\ \hspace{1em}\subset \mathbb{P}(2,3^2,4,5^2,6,7,8)\hfill \end{array}$	$3\times \frac{1}{2}(1,1,1),5\times \frac{1}{3}(1,2,2),\frac{1}{5}(2,3,4),\frac{1}{8}(3,5,7)$	$\frac{11}{120}$	1	$\frac{781}{120}$	$\begin{array}{c}2\\ (0,1,1,2,3)\end{array}$	$\begin{array}{c}2\\ 9,8,8,7,6\end{array}$	$\begin{array}{cccc}3& 3& 4& 5\\ & 4& 5& 6\\ & & 5& 6\\ & & & 7\end{array}$
$\begin{array}{c}{X}_{10,{11}^2,{12}^4,{13}^2,14}\hfill \\ \hspace{1em}\subset \mathbb{P}(3,4^2,5^2,6^2,7^2)\hfill \end{array}$	$3\times \frac{1}{2}(1,1,1),2\times \frac{1}{5}(1,4,4),2\times \frac{1}{7}(3,4,6)$	$\frac{3}{70}$	1	$\frac{267}{70}$	$\begin{array}{c}4\\ (0,0,1,1,2)\end{array}$	$\begin{array}{c}4\\ 8,8,7,7,6\end{array}$	$\begin{array}{cccc}4& 5& 5& 6\\ & 5& 5& 6\\ & & 6& 7\\ & & & 7\end{array}$
$\begin{array}{c}{X}_{12,13,{14}^2,{15}^2,{16}^2,17,18}\hfill \\ \hspace{1em}\subset \mathbb{P}(3,4,5,6,7^2,8,9,10)\hfill \end{array}$	$\frac{1}{2}(1,1,1),\frac{1}{4}(1,3,3),\frac{1}{5}(2,3,4),\frac{1}{7}(3,4,6),\frac{1}{10}(3,7,9)$	$\frac{3}{140}$	1	$\frac{393}{140}$	$\begin{array}{c}4\\ (0,1,1,2,3)\end{array}$	$\begin{array}{c}4\\ 11,10,10,9,8\end{array}$	$\begin{array}{cccc}5& 5& 6& 7\\ & 6& 7& 8\\ & & 7& 8\\ & & & 9\end{array}$
$\begin{array}{c}{X}_{12,13,14,15,{16}^2,17,18,19,20}\hfill \\ \hspace{1em}\subset \mathbb{P}(3,4,5,6,7,8,9,10,11)\hfill \end{array}$	$2\times \frac{1}{2}(1,1,1),3\times \frac{1}{3}(1,2,2),\frac{1}{5}(1,4,4),\frac{1}{11}(4,7,10)$	$\frac{1}{55}$	1	$\frac{149}{55}$	$\begin{array}{c}3\\ (0,1,2,3,4)\end{array}$	$\begin{array}{c}3\\ 13,12,11,10,9\end{array}$	$\begin{array}{cccc}4& 5& 6& 7\\ & 6& 7& 8\\ & & 8& 9\\ & & & 10\end{array}$

[Kreuzer and Skarke 00] M. Kreuzer and H. Skarke. “ Complete Classification of Reflexive Polyhedra in Four Dimensions.” Adv. Theor. Math. Phys., 4:6 (2000), 1209–1230.

 Web of Science ®Google Scholar

[Iskovskikh and Prokhorov 99] V. A. Iskovskikh and Yu. G. Prokhorov. “Fano Varieties.” In Algebraic Geometry, V, vol. 47, Encyclopaedia Math. Sci., pp. 1–247. Berlin: Springer, 1999.

 Google Scholar

[Altınok 05] S. Altınok. “Constructing New K3 Surfaces.” Turkish J. Math., 29:2 (2005), 175–192.

 Google Scholar