Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Views on level $\mathit \ell$ curves, K3 surfaces and Fano threefolds   An analogue of the Mukai map $m_g: \mathcal P_g \to \mathcal M_g$ is studied
for the moduli $\mathcal R_{g, \ell}$ of genus $g$ curves $C$ with a level
$\ell$ structure. Let $\mathcal P^{\perp}_{g, \ell}$ be the moduli space of
$4$-tuples $(S, \mathcal L, \mathcal E, C)$ so that $(S, \mathcal L)$ is a
polarized K3 surface of genus $g$, $\mathcal E$ is orthogonal to $\mathcal L$
in Pic$S$ and defines a standard degree $\ell$ K3 cyclic cover of $S$, $C \in
\vert \mathcal L \vert$. We say that $(S, \mathcal L, \mathcal E)$ is a level
$\ell$ K3 surface. These exist for $\ell \leq 8$ and their families are known.
We define a level $\ell$ Mukai map $r_{g, \ell}: \mathcal P^{\perp}_{g, \ell}
\to \mathcal R_{g, \ell}$, induced by the assignment of $(S, \mathcal L,
\mathcal E, C)$ to $ (C, \mathcal E \otimes \mathcal O_C)$. We investigate a
curious possible analogy between $m_g$ and $r_{g, \ell}$, that is, the failure
of the maximal rank of $r_{g, \ell}$ for $g = g_{\ell} \pm 1$, where $g_{\ell}$
is the value of $g$ such that $\dim \mathcal P^{\perp}_{g, \ell} = \dim
\mathcal R_{g,\ell}$. This is proven here for $\ell = 3$. As a related open
problem we discuss Fano threefolds whose hyperplane sections are level $\ell$
K3 surfaces and their classification.

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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: supr-con
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The categories of this paper are: cs
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat