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In this paper, we study relations between various natural structures on F-manifolds. In particular, given an arbitrary Riemannian F-manifold, we present a construction of a canonical flat F-manifold associated to it. We also describe a construction of a canonical homogeneous Riemannian F-manifold associated to an arbitrary exact homogeneous flat pencil of metrics satisfying a certain non-degeneracy assumption. In the last part of the paper, we construct Legendre transformations for Riemannian F-manifolds. [ABSTRACT FROM AUTHOR]

Let |$G$| be a complex connected simple algebraic group with a fixed real form |$\sigma $|⁠. Let |$G({\mathbb{R}})=G^\sigma $| be the corresponding group of real points. This paper reports a finiteness theorem for the classification of irreducible unitary Harish-Chandra modules of |$G({\mathbb{R}})$| (up to equivalence) having nonvanishing Dirac cohomology. Moreover, we study the distribution of the spin norm along Vogan pencils for certain |$G({\mathbb{R}})$|⁠ , with particular attention paid to the unitarily small convex hull introduced by Salamanca-Riba and Vogan. [ABSTRACT FROM AUTHOR]

The flex divisor |$R_{\textrm flex}$| of a primitively polarized K3 surface |$(X,L)$| is, generically, the set of all points |$x\in X$| for which there exists a pencil |$V\subset |L|$| whose base locus is |$\{x\}$|⁠. We show that if |$L^2=2d$| then |$R_{\textrm flex}\in |n_dL|$| with $$ \begin{align*} &n_d= \frac{(2d)!(2d+1)!}{d!^2(d+1)!^2} =(2d+1)C(d)^2,\end{align*}$$ where |$C(d)$| is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space |$F_{2d}$| of polarized K3 surfaces. [ABSTRACT FROM AUTHOR]

In the moduli space ${{\mathcal {R}}}_g$ of double étale covers of curves of a fixed genus $g$ , the locus of covers of curves with a semicanonical pencil decomposes as the union of two divisors— ${{\mathcal {T}}}^e_g$ and ${{\mathcal {T}}}^o_g$. Adapting arguments of Teixidor for the divisor of curves having a semicanonical pencil, we prove that both divisors are irreducible and compute their divisor classes in the Deligne–Mumford compactification ${\overline {{\mathcal {R}}}}_g$. [ABSTRACT FROM AUTHOR]

We show |$\overline{\mathcal{M}}_{10, 10}$| and |$\overline{\mathcal{F}}_{11,9}$| have Kodaira dimension zero. Our method relies on the construction of a number of curves via nodal Lefschetz pencils on blown-up |$K3$| surfaces. The construction further yields that any effective divisor in |$\overline{\mathcal{M}}_{g}$| with slope |$<6+(12-\delta)/(g+1)$| must contain the locus of curves that are the normalization of a |$\delta $| -nodal curve lying on a |$K3$| surface of genus |$g+\delta $|⁠. [ABSTRACT FROM AUTHOR]

Consider a one-parameter family of smooth, irreducible, projective curves of genus |$g\ge 2$| defined over a number field. Each fiber contains at most finitely many rational points by the Mordell conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell–Weil rank of the fiber's Jacobian. Our proof uses Vojta's approach to the Mordell Conjecture furnished with a height inequality due to the 2nd- and 3rd-named authors. In addition we obtain uniform bounds for the number of torsion points in the Jacobian that lie in each fiber of the family. [ABSTRACT FROM AUTHOR]

Let f(x,y)=0 and l(x,y)=0 be, respectively, a singular and a regular analytic curve defined in the neighborhood of the origin of the complex plane. We study the family of analytic curves f(x,y)−tl(x,y)M=0, where t≠0 is a complex parameter. All curves of this family but a finite number have the same embedded topological type. We show that the number of exceptions does not exceed the number of components of the curve f(x,y)=0 counted without multiplicities. Then we apply this result to estimate the number of critical values at infinity of complex polynomials in two variables. [ABSTRACT FROM AUTHOR]