# 38406501359372282063949 & all that: Monodromy of Fano Problems

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A Fano problem is an enumerative problem of counting $r$-dimensional linear subspaces on a complete intersection in $\mathbb{P}^n$ over a field of arbitrary characteristic, whenever the corresponding Fano scheme is finite. A classical example is enumerating lines on a cubic surface. We study the monodromy of finite Fano schemes $F_{r}(X)$ as the complete intersection $X$ varies. We prove that the monodromy group is either symmetric or alternating in most cases. In the exceptional cases, the monodromy group is one of the Weyl groups $W(E_6)$ or $W(D_k)$.

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- 38406501359372282063949allthatMonodromyofFanoProblems
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- article
- date_added
- 2020-05-06
- date_published
- 2020-10-09

### BibTeX entry

@article{38406501359372282063949allthatMonodromyofFanoProblems, key = {38406501359372282063949allthatMonodromyofFanoProblems}, type = {article}, title = {38406501359372282063949 {\&} all that: Monodromy of Fano Problems}, author = {Sachi Hashimoto and Borys Kadets}, abstract = {A Fano problem is an enumerative problem of counting {\$}r{\$}-dimensional linear subspaces on a complete intersection in {\$}\mathbb{\{}P{\}}^n{\$} over a field of arbitrary characteristic, whenever the corresponding Fano scheme is finite. A classical example is enumerating lines on a cubic surface. We study the monodromy of finite Fano schemes {\$}F{\_}{\{}r{\}}(X){\$} as the complete intersection {\$}X{\$} varies. We prove that the monodromy group is either symmetric or alternating in most cases. In the exceptional cases, the monodromy group is one of the Weyl groups {\$}W(E{\_}6){\$} or {\$}W(D{\_}k){\$}.}, comment = {}, date_added = {2020-05-06}, date_published = {2020-10-09}, urls = {http://arxiv.org/abs/2002.04580v1,http://arxiv.org/pdf/2002.04580v1}, collections = {attention-grabbing-titles,combinatorics,integerology}, url = {http://arxiv.org/abs/2002.04580v1 http://arxiv.org/pdf/2002.04580v1}, year = 2020, urldate = {2020-05-06}, archivePrefix = {arXiv}, eprint = {2002.04580}, primaryClass = {math.AG} }