We consider the family of cyclic fourfold covers $w^4 = \prod_{j=1}^{7}(z-x_j)$ of the complex projective line branched at the eight points $x_1,\ldots,x_7,\infty$. The period map identifies the configuration space $X(2,8)$ of the branch points with a Zariski open subset of a quotient of the five-dimensional complex ball. The inverse of the period map is expressed projectively by $105$ automorphic forms $f_J$, which are proportional to the signed branch-point polynomials $\hat x_J$ with a common scalar factor. We determine this factor for the period $\eta$ of the differential $dz/w$: it is the product of the constant $-1/(2^{12}\Gamma(3/4)^{16})$ and the square of a quadratic form in $\eta$.
Let $k$ be an algebraically closed field of characteristic zero. Degree-four rational maps on $\mathbb{P}^1$, up to conjugation, correspond to pairs of binary forms $(F,G)\in V_5\oplus V_3$. The associated invariant ring is the joint invariant ring $\mathcal{R}_{5,3}=k[V_5\oplus V_3]^{\mathrm{SL}_2}$. We determine a mi...
We study the rational points on the star curve $X_0^*(N) := X_0(N)/W(N)$, the quotient of the classical modular curve $X_0(N)$ by the full group of Atkin--Lehner involutions, for squarefree levels $N$. Rational points on $X_0^*(N)$ parameterize $\mathbb{Q}$-curves, i.e.\ elliptic curves $E/\overline{\mathbb{Q}}$ that a...
Eran Assaf, Sachi Hashimoto, A. Shnidman· 0 citations
Let $C\subset X(1)\times X(1)$ be a smooth curve defined over $\mathbb{C}_p$ with irreducible reduction. We study the intersection loci of $C$ with the modular subvarieties $Y_0(n)$. If the curve $C$ avoids points where both coordinates have supersingular reduction, or if a sequence $Y_0(a_n)$ is taken where the number...
In this paper, over the field of complex numbers, we prove that a numerical Godeaux surface contains at most six pairwise disjoint $(-2)$-curves, and that this bound is sharp. As an application, we refine the classification of involutions on smooth minimal surfaces of general type with $p_g=0$ and $K^2=7$: the divisori...
We prove that the integral cycle class map for $\SP^d(\PP^n)$ is split injective, with finite cokernel annihilated by $d!$,and we determine the ranks of Chow groups for $\mathrm{SP}^d(\mathbb{P}^n)$. Compatible integral transfer retractions show that $\CH_q(\SP^d(\PP^n))$ stabilizes for $d\geq q$. An explicit nonalgebr...
Finiteness and injectivity of the moduli morphisms $\mu_n\colon|nH|_{\rm{sm}}\to\mathcal{M}_{g_n}$ associated with linear systems on K3 surfaces are understood in many cases. In this paper we study their ramification. Existing vanishing and stability results already imply unramifiedness when the polarisation or the mul...
Dario Faro, Frank Gounelas· 0 citations
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