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HLM22 Smooth Fano fourfolds of Picard number two

Hausen, J., Laface, A., & Mauz, C. (2022). On smooth Fano fourfolds of Picard number two. Rev. Mat. Iberoam., 38(1), 53–93.

count67 families
constructioncox-ring-hypersurface
Picard rank $\rho_X$2
entry numberingTable 1 row (No.)
invariantshodge, volume, hilbert_series, contractions
links10.4171/rmi/1271 MR4382464
notes67 smooth Fano fourfolds of Picard number two with a general hypersurface Cox ring (Theorem 1.1). Elementary contractions (Proposition 6.2), Hodge numbers (Propositions 7.1–7.2), volumes and Hilbert series provided.

Varieties

Each row is a family from this source. The description column records the construction as given in the paper (ambient and defining bundle, Cox-ring data, base variety, or birational model, depending on the source); the invariant columns are those the paper tabulates. Provenance and conventions are noted below the table.

How to read the construction. Each family is a general hypersurface in a toric variety of Picard number two. Its Cox ring is a polynomial ring in seven variables graded by the divisor class group $\mathbb{Z}^2$; the two rows of the matrix $w$ are the $\mathbb{Z}^2$-degrees of those seven generators. The fourfold $X$ is cut out by a general polynomial $g$ of bidegree $\deg(g)$, and $-K$ is its anticanonical class in $\mathbb{Z}^2$; the volume is $(-K)^4$. In short: $X=\{g=0\}$ inside the toric variety whose homogeneous coordinates carry the degrees $w$.

entry$\rho_X$$(-\mathrm{K}_X)^4$$\mathrm{h}^{1,1}$$\mathrm{h}^{1,2}$$\mathrm{h}^{1,3}$$\mathrm{h}^{2,2}$description
HLM22:124322003w=[[1,1,1,1,0,0,0],[0,0,0,0,1,1,1]]; deg(g)=(1,1); -K=(3,2)
HLM22:2225620010w=[[1,1,1,1,0,0,0],[0,0,0,0,1,1,1]]; deg(g)=(2,1); -K=(2,2)
HLM22:328020029w=[[1,1,1,1,0,0,0],[0,0,0,0,1,1,1]]; deg(g)=(3,1); -K=(1,2)
HLM22:422702003w=[[1,1,1,1,0,0,0],[0,0,0,0,1,1,1]]; deg(g)=(1,2); -K=(3,1)
HLM22:5211220340w=[[1,1,1,1,0,0,0],[0,0,0,0,1,1,1]]; deg(g)=(2,2); -K=(2,1)
HLM22:62262030185w=[[1,1,1,1,0,0,0],[0,0,0,0,1,1,1]]; deg(g)=(3,2); -K=(1,1)
HLM22:724162004w=[[1,1,1,1,0,0,-1],[0,0,0,0,1,1,1]]; deg(g)=(1,1); -K=(2,2)
HLM22:8216320123w=[[1,1,1,1,0,0,-1],[0,0,0,0,1,1,1]]; deg(g)=(1,2); -K=(2,1)
HLM22:9222420014w=[[1,1,1,1,0,0,-1],[0,0,0,0,1,1,1]]; deg(g)=(2,1); -K=(1,2)
HLM22:102522018126w=[[1,1,1,1,0,0,-1],[0,0,0,0,1,1,1]]; deg(g)=(2,2); -K=(1,1)
HLM22:1124642005w=[[1,1,1,1,0,0,-2],[0,0,0,0,1,1,1]]; deg(g)=(1,1); -K=(1,2)
HLM22:12298201295w=[[1,1,1,1,0,0,-2],[0,0,0,0,1,1,1]]; deg(g)=(1,2); -K=(1,1)
HLM22:1323522004w=[[1,1,1,1,0,0,0],[0,0,0,1,1,1,1]]; deg(g)=(1,2); -K=(3,2)
HLM22:1426520665w=[[1,1,1,1,0,0,0],[0,0,0,1,1,1,1]]; deg(g)=(2,3); -K=(2,1)
HLM22:1528320555w=[[1,1,1,1,0,0,-1],[0,0,0,1,1,1,1]]; deg(g)=(1,3); -K=(2,1)
HLM22:1623522006w=[[1,1,1,1,1,0,0],[0,0,0,0,1,1,1]]; deg(g)=(2,1); -K=(3,2)
HLM22:1728120977w=[[1,1,1,1,1,0,0],[0,0,0,0,1,1,1]]; deg(g)=(3,2); -K=(2,1)
HLM22:182382021143w=[[1,1,1,1,1,0,0],[0,0,0,0,1,1,1]]; deg(g)=(3,1); -K=(1,1)
HLM22:19219220122w=[[1,1,1,1,0,0,0],[-1,0,0,0,1,1,1]]; deg(g)=(2,1); -K=(2,1)
HLM22:2024322003w=[[1,1,1,1,0,0,0],[-1,0,0,0,1,1,1]]; deg(g)=(1,1); -K=(3,1)
HLM22:21211320553w=[[1,1,1,1,1,0,0],[-1,0,0,0,1,1,1]]; deg(g)=(3,1); -K=(2,1)
HLM22:22227220010w=[[1,1,1,1,0,0,0],[0,0,1,1,1,1,1]]; deg(g)=(2,2); -K=(2,3)
HLM22:232512013103w=[[1,1,1,1,0,0,0],[0,0,1,1,1,1,1]]; deg(g)=(3,3); -K=(1,2)
HLM22:242342035218w=[[1,1,1,2,0,0,0],[0,0,1,2,1,1,1]]; deg(g)=(4,4); -K=(1,2)
HLM22:2521720114591w=[[1,1,2,3,0,0,0],[0,0,2,3,1,1,1]]; deg(g)=(6,6); -K=(1,2)
HLM22:26221620010w=[[1,1,1,0,0,0,0],[0,0,1,1,1,1,1]]; deg(g)=(2,2); -K=(1,3)
HLM22:272642020138w=[[1,1,1,0,0,0,0],[0,0,2,1,1,1,1]]; deg(g)=(2,4); -K=(1,2)
HLM22:282820112570w=[[1,1,1,0,0,0,0],[0,0,3,1,1,1,1]]; deg(g)=(2,6); -K=(1,1)
HLM22:29219220122w=[[1,1,1,1,0,0,0],[0,0,0,1,1,1,1]]; deg(g)=(2,2); -K=(2,2)
HLM22:302182045255w=[[1,1,1,1,0,0,0],[0,0,0,1,1,1,1]]; deg(g)=(3,3); -K=(1,1)
HLM22:31248201094w=[[1,1,1,2,0,0,0],[0,0,0,1,1,1,1]]; deg(g)=(4,2); -K=(1,2)
HLM22:3221220100508w=[[1,1,1,2,0,0,0],[0,0,0,2,1,1,1]]; deg(g)=(4,4); -K=(1,1)
HLM22:332502024162w=[[1,1,2,1,0,0,0],[0,1,3,2,1,1,1]]; deg(g)=(4,6); -K=(1,3)
HLM22:3423782004w=[[1,1,1,1,1,0,0],[0,1,1,1,1,1,1]]; deg(g)=(2,2); -K=(3,4)
HLM22:35214420128w=[[1,1,1,1,1,0,0],[0,1,1,1,1,1,1]]; deg(g)=(3,3); -K=(2,3)
HLM22:362202022162w=[[1,1,1,1,1,0,0],[0,1,1,1,1,1,1]]; deg(g)=(4,4); -K=(1,2)
HLM22:3729620560w=[[1,1,1,1,2,0,0],[0,1,1,1,2,1,1]]; deg(g)=(4,4); -K=(2,3)
HLM22:382102071402w=[[1,1,1,1,3,0,0],[0,1,1,1,3,1,1]]; deg(g)=(6,6); -K=(1,2)
HLM22:392482024170w=[[1,1,1,2,3,0,0],[0,1,1,2,3,1,1]]; deg(g)=(6,6); -K=(2,3)
HLM22:4023522004w=[[1,1,1,1,0,0,0],[0,1,1,1,1,1,1]]; deg(g)=(2,2); -K=(2,4)
HLM22:4129921123w=[[1,1,1,1,0,0,0],[0,1,1,1,1,1,1]]; deg(g)=(3,3); -K=(1,3)
HLM22:42230420010w=[[1,1,1,1,0,0,0],[0,2,2,2,1,1,1]]; deg(g)=(2,4); -K=(2,5)
HLM22:432542119131w=[[1,1,1,1,0,0,0],[0,2,2,2,1,1,1]]; deg(g)=(3,6); -K=(1,3)
HLM22:4426621554w=[[1,1,1,2,0,0,0],[0,1,1,2,1,1,1]]; deg(g)=(4,4); -K=(1,3)
HLM22:452362150288w=[[1,1,1,2,0,0,0],[0,2,2,4,1,1,1]]; deg(g)=(4,8); -K=(1,3)
HLM22:462332124163w=[[1,1,2,3,0,0,0],[0,1,2,3,1,1,1]]; deg(g)=(6,6); -K=(1,3)
HLM22:4721821159793w=[[1,1,2,3,0,0,0],[0,2,4,6,1,1,1]]; deg(g)=(6,12); -K=(1,3)
HLM22:4824332003w=[[1,1,1,1,1,0,0],[0,1,1,1,2,1,1]]; deg(g)=(2,2); -K=(3,5)
HLM22:49214521231w=[[1,1,1,1,1,0,0],[0,2,2,2,3,1,1]]; deg(g)=(3,6); -K=(2,5)
HLM22:50214420340w=[[1,1,1,1,0,0,0],[0,1,1,2,1,1,1]]; deg(g)=(2,4); -K=(2,3)
HLM22:512222065356w=[[1,1,1,2,0,0,0],[0,1,1,3,1,1,1]]; deg(g)=(4,6); -K=(1,2)
HLM22:522652020139w=[[1,1,1,2,1,0,0],[0,1,1,3,2,1,1]]; deg(g)=(4,6); -K=(2,3)
HLM22:5324312003w=[[1,1,1,1,1,1,0],[-1,0,0,0,0,1,1]]; deg(g)=(2,0); -K=(4,1)
HLM22:5426220672w=[[1,1,1,1,1,1,0],[-1,0,0,0,0,1,1]]; deg(g)=(4,0); -K=(2,1)
HLM22:5523762008w=[[1,1,1,1,1,2,0],[-1,0,0,0,0,1,1]]; deg(g)=(3,0); -K=(4,1)
HLM22:56234120121w=[[1,1,1,1,1,3,0],[-1,0,0,0,0,1,1]]; deg(g)=(4,0); -K=(4,1)
HLM22:572312025181w=[[1,1,1,1,3,1,0],[-1,0,0,0,0,1,1]]; deg(g)=(6,0); -K=(2,1)
HLM22:5821625202w=[[1,1,1,1,3,0,0],[0,0,0,0,0,1,1]]; deg(g)=(6,0); -K=(1,2)
HLM22:5926422102w=[[1,1,1,2,3,0,0],[0,0,0,0,0,1,1]]; deg(g)=(6,0); -K=(2,2)
HLM22:6028022102w=[[1,1,1,2,3,1,0],[0,0,0,0,0,1,1]]; deg(g)=(6,0); -K=(3,2)
HLM22:61212821002w=[[1,1,1,1,2,0,0],[0,0,0,0,0,1,1]]; deg(g)=(4,0); -K=(2,2)
HLM22:62216021002w=[[1,1,1,1,2,1,0],[0,0,0,0,0,1,1]]; deg(g)=(4,0); -K=(3,2)
HLM22:6321922502w=[[1,1,1,1,1,0,0],[0,0,0,0,0,1,1]]; deg(g)=(3,0); -K=(2,2)
HLM22:6422402502w=[[1,1,1,1,1,1,0],[0,0,0,0,0,1,1]]; deg(g)=(3,0); -K=(3,2)
HLM22:6524322002w=[[1,1,1,1,1,0,0],[0,0,0,0,0,1,1]]; deg(g)=(2,0); -K=(3,2)
HLM22:6624802002w=[[1,1,1,1,1,1,0],[0,0,0,0,0,1,1]]; deg(g)=(2,0); -K=(4,2)
HLM22:6726242002w=[[1,1,1,1,1,2,0],[0,0,0,0,0,1,1]]; deg(g)=(2,0); -K=(5,2)

Volumes from Theorem 1.1; Hodge numbers from Proposition 7.2. Numbering follows the paper's Table 1 (No. 1-67). Source: arXiv:1907.08000.