bio  website  math.sunysb.edu/~jstarr/… 

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17h

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Moduli of curves in characteristic zero
You are using nonstandard notations. The notation $\overline{M}_{g,n}(K)$ usually denotes the set of $K$valued points of the scheme $\overline{M}_{g,n}$, not a (finite type) $K$scheme. You could ask, is the natural morphism of $\overline{K}$schemes, $$\overline{M}_{g,n,\overline{K}} \to \overline{M}_{g,n,K}\times_{\text{Spec}(K)} \text{Spec}(\overline{K})$$ an isomorphism? The answer to this is yes, because the formation of the coarse moduli space is compatible with flat base change. I recommend reading KeelMori for a detailed discussion of this property. 
1d

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Lifting to char 0, references and questions
I forgot to click community wiki :( I clicked it now. 
1d

answered  Lifting to char 0, references and questions 
2d

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rank of Abelian schemes under ample hypersurface section
@TimoKeller: Yes, I think you need uncountability for all of the arguments in that paper (unfortunately). 
Jul 7 
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rank of Abelian schemes under ample hypersurface section
@TimoKeller For the counterexample, I think there are no problems with $k$ being countable. However, you are quite correct that, in the last paragraph, "sufficiently general" almost certainly does require that $k$ is uncountable. 
Jul 7 
answered  rank of Abelian schemes under ample hypersurface section 
Jul 4 
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A construction of the HilbertChow morphism
Matthieu's suggestion is excellent. Another reference is Section I.6 of Koll'ar's "Rational Curves on Algebraic Varieties". 
Jul 3 
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closed form expression for an infinite series
@Lucian: Of course I have no idea what the OP meant, but I thought that I would mention that my comment is still valid if the factor is $q^{n(n+1)/2}$. After a change of variables $s=1/q$, my comment shows that the expression cannot be a rational function in $s$ and $u$. But every rational function in $q$ and $u$ is also a rational function in $s$ and $u$. 
Jul 3 
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Two questions about counter example for Torelli theorem for hyperkahler manifolds
Regarding (1),it looks to me like $\Sigma$ is the image of an embedding $T\to \overline{K}^2(T)$ sending $p$ to $2(\underline{p}) + \underline{2p}$. Also, $F$ seems to be bimeromorphic to a projective space bundle over $\Sigma$. Since the Albanese variety is a bimeromorphic invariant, that would imply that the map from $F$ to $\Sigma$ is (equivalent to) the Albanese morphism for $F$. 
Jul 3 
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closed form expression for an infinite series
Since the coefficient of $u^n$ is a polynomial in $q$ of degree, roughly, $n^2$, there should be no closed form expression that is a rational function. Since the coefficients of $u^n$ satisfy a linear recurrence, the $q$degrees would have to grow linearly in $n$. Maybe it could have a closed expression that is a rational function in $u$, but the coefficients are not all polynomials in $q$. 
Jul 3 
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A generalization of the Grauert direct image theorem
Your statement is wrong. The pushforward $f_*\mathcal{F}$ might be identically zero, yet $H^0(X_y,\mathcal{F}_y)$ nonzero for some points $y$. 
Jul 1 
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Flat family of Hilbert scheme of points
The reference for the Douady spaces is the following. MR0203082 (34 #2940) Reviewed. Douady, Adrien. Le problème des modules pour les sousespaces analytiques compacts d'un espace analytique donné. (French) Ann. Inst. Fourier (Grenoble) 16 1966 fasc. 1, 1–95. 32.47 (57.70) 
Jul 1 
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Flat family of Hilbert scheme of points
The reference in the algebraic case is Chapitre IV of the following. MR0146040 (26 #3566) Reviewed. Grothendieck, Alexander. Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.] (French) Secrétariat mathématique, Paris 1962 ii+205 pp. 14.00 
Jul 1 
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Flat family of Hilbert scheme of points
Is your family algebraic? If so, this follows from Grothendieck's original construction of the Hilbert scheme  which was in the relative setting. If your family is Kaehlerian, but not necessarily algebraic, you should look up "Douady spaces". 
Jun 30 
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smooth affine surfaces over algebraically closed fields with trivial ltorsion of the Brauer group
@DanielLoughran: If the characteristic is $p$, then there are actions on $\mathbb{A}^2_k$ of $p$groups such that the quotient map is finite 'etale. For instance, translations have order $p$. 
Jun 30 
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Surjectivity of certain cohomology groups on hypersurfaces of high degree
Speaking for myself, thank you for clarifying the question. I was having trouble understanding how the OP's formulation could be true. 
Jun 26 
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Vanishing theorems for pluricanonical bundle
No, certainly not. Consider the case that $X$ is Fano, e.g., $\mathbb{P}^r$. 
Jun 26 
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complementary bundle for a divisor
I do not understand the question. Are you trying to split the fundamental exact sequence $$\mathcal{I}/\mathcal{I}^2 \to \Omega_X_D \to \Omega_D \to 0,$$ that is associated to a divisor $D$ inside a complex manifold $X$, with ideal sheaf $\mathcal{I}$? Or are you trying to decompose $\mathcal{I}/\mathcal{I}^2$ as a direct sum of holomorphic invertible sheaves? 
Jun 25 
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Toric morphism fiber and kernel dimensions
In fact, even if the morphism is torus equivariant, and it is an isomorphism on the open orbit, this can still fail. Just consider the iterated blowup of the affine plane, first at the origin, then at one of the two torus invariant points on the exceptional divisor. The fiber of this new scheme over the origin in the plane is a union of two $\mathbb{P}^1$s, and the condition fails at the intersection point. 
Jun 25 
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Transcendental numbers in the padic rationals $\mathbb Q_p$
What is your question supposed to mean? By definition, if $E/K$ and $F/K$ are both purely transcendental with transcendence degree $1$, then $E$ is isomorphic to $F$ as a $K$extension. 