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Questions about K3 surfaces, which are smooth complex surfaces $X$ with trivial canonical bundle and vanishing $H^1(O_X)$. They are examples of Calabi-Yau varieties of dimension $2$.

5 votes

Period integrals of the fiber of elliptically fibered K3 manifolds

I figure that there are better answers possible, but this might do for a stater: Assuming that your $f$ and $g$ are minimimal (i.e. there is no $u$, with $\deg(u)>0$ s.t $u^4$ divides $f$ and $u^6$ …
Remke Kloosterman's user avatar
3 votes
Accepted

Neron-Severi Lattice of Elliptic K3

Usually, the really hard part of this problem is to find a basis for the Mordell-Weil group, not to calculate the N\'eron-Severi group. You suggest that somehow you are capable to find a basis for the …
Remke Kloosterman's user avatar
3 votes

Polarizations of K3 surfaces over finite fields

Let $S$ be a smooth projective surface and $m:=\gcd(\deg \lambda)$ where $\lambda$ runs through the ample cone. Let $m'$ be the gcd of the entries of the intersection matrix of $S$ Then $m$ equals ei …
Remke Kloosterman's user avatar
4 votes
Accepted

Question on K3 Surface

I slightly disagree with Tony's answer: If you want to obtain a smooth K3 surface you need to pick a ramification divisor that is smooth and is linear equivalent to twice the anti-canonical class. I …
Remke Kloosterman's user avatar
1 vote
Accepted

Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory

Question 1: $S'$ is an elliptic surface without a section such that it is Jacobian is $S$. The surfaces with this property are parametrized by a certain cohomology group and the cocycle the authors re …
Remke Kloosterman's user avatar
10 votes

Does $(x^2 - 1)(y^2 - 1) = c z^4$ have a rational point, with z non-zero, for any given rati...

If you take the projective closure of your surface in $\mathbb{P}^3$ you find a singular quartic. This quartic has six singular points. Namely the two points found by Daniel and four points of the for …
Remke Kloosterman's user avatar