The blow-up of the cubic fourfold along eight planes is Fano

Abstract

We prove that the blow-up Z_m of a smooth cubic fourfold along m <= 8 planes, pairwise meeting at distinct points and satisfying explicit generality conditions, is a smooth Fano fourfold of Picard rank rho(Z_m) = m+1 <= 9; in particular, we obtain three new families of smooth Fano fourfolds with rho > 6 that are not a product of surfaces. Complementary to this result, we show that a smooth cubic fourfold as above exists. Moreover, we study the birational geometry of Z_m and show that it corresponds exactly to that of a smooth del Pezzo surfaces S_m, obtained as the blow-up of P^2 along m <= 8 points in general position. Namely, Z_m and S_m have isomorphic Mori cones, pseudo-effective cones with same Mori chamber decomposition, and there is a one-to-one correspondence between the contractions of Z_m and the contractions of S_m. Lastly, we study the anticanonical linear system of Z_m and prove that it is basepoint free if and only if m <= 7. For m = 8 and a special configuration of planes, we show that the anticanonical base locus Bs(|-K_{Z_8}|) has dimension 2 and contains a smooth del Pezzo surface of degree 1.

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BibTeX RIS

BibTeX

@misc{secci2026blowup,
  title = {{The blow-up of the cubic fourfold along eight planes is Fano}},
  author = {Saverio Andrea Secci},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25532},
  eprint = {2610.25532},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25532},
}

RIS

TY  - UNPB
TI  - The blow-up of the cubic fourfold along eight planes is Fano
AU  - Saverio Andrea Secci
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25532
UR  - https://arxiv.org/abs/2610.25532
AB  - We prove that the blow-up Z_m of a smooth cubic fourfold along m <= 8 planes, pairwise meeting at distinct points and satisfying explicit generality conditions, is a smooth Fano fourfold of Picard rank rho(Z_m) = m+1 <= 9; in particular, we obtain three new families of smooth Fano fourfolds with rho > 6 that are not a product of surfaces. Complementary to this result, we show that a smooth cubic fourfold as above exists. Moreover, we study the birational geometry of Z_m and show that it corresponds exactly to that of a smooth del Pezzo surfaces S_m, obtained as the blow-up of P^2 along m <= 8 points in general position. Namely, Z_m and S_m have isomorphic Mori cones, pseudo-effective cones with same Mori chamber decomposition, and there is a one-to-one correspondence between the contractions of Z_m and the contractions of S_m. Lastly, we study the anticanonical linear system of Z_m and prove that it is basepoint free if and only if m <= 7. For m = 8 and a special configuration of planes, we show that the anticanonical base locus Bs(|-K_{Z_8}|) has dimension 2 and contains a smooth del Pezzo surface of degree 1.
ER  -

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