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Let $Z$ be a closed subscheme of a smooth complex projective variety $Ysubseteq Ps^N$, with $dim,Y=2r+1geq 3$. We describe the intermediate Neron-Severi group (i.e. the image of the cycle map $A_r(X)to H_{2r}(X;mathbb{Z})$) of a general smooth hypersurface $Xsubset Y$ of sufficiently large degree containing $Z$.
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The Noether-Lefschetz theorem asserts that any curve in a very general surface $X$ in $mathbb P^3$ of degree $d geq 4$ is a restriction of a surface in the ambient space, that is, the Picard number of $X$ is $1$. We proved previously that under some conditions, which replace the condition $d geq 4$, a very general surface in a simplicial toric threefold $mathbb P_Sigma$ (with orbifold singularities) has the same Picard number as $mathbb P_Sigma$. Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in $mathbb P_Sigma$ in a linear system of a Cartier ample divisor with respect to a (-1)-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether-Lefschetz loci which contain a line, defined as a rational curve that is minimal in a suitable sense.
Noether-Lefschetz divisors in the moduli of K3 surfaces are the loci corresponding to Picard rank at least 2. We relate the degrees of the Noether-Lefschetz divisors in 1-parameter families of K3 surfaces to the Gromov-Witten theory of the 3-fold total space. The reduced K3 theory and the Yau-Zaslow formula play an important role. We use results of Borcherds and Kudla-Millson for O(2,19) lattices to determine the Noether-Lefschetz degrees in classical families of K3 surfaces of degrees 2, 4, 6 and 8. For the quartic K3 surfaces, the Noether-Lefschetz degrees are proven to be the Fourier coefficients of an explicitly computed modular form of weight 21/2 and level 8. The interplay with mirror symmetry is discussed. We close with a conjecture on the Picard ranks of moduli spaces of K3 surfaces.
Let $Z$ be a closed subscheme of a smooth complex projective complete intersection variety $Ysubseteq Ps^N$, with $dim Y=2r+1geq 3$. We describe the Neron-Severi group $NS_r(X)$ of a general smooth hypersurface $Xsubset Y$ of sufficiently large degree containing $Z$.
The Yau-Zaslow conjecture determines the reduced genus 0 Gromov-Witten invariants of K3 surfaces in terms of the Dedekind eta function. Classical intersections of curves in the moduli of K3 surfaces with Noether-Lefschetz divisors are related to 3-fold Gromov-Witten theory via the K3 invariants. Results by Borcherds and Kudla-Millson determine the classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise. Via a detailed study of the STU model (determining special curves in the moduli of K3 surfaces), we prove the Yau-Zaslow conjecture for all curve classes on K3 surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper.
We consider the Noether-Lefschetz problem for surfaces in Q-factorial normal 3-folds with rational singularities. We show the existence of components of the Noether-Lefschetz locus of maximal codimension, and that there are indeed infinitely many of them. Moreover, we show that their union is dense in the natural topology.
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