Rational curves on Fermat hypersurfaces

In this note we study rational curves on degree $p^r+1$ Fermat hypersurface in \PP^{p^r+1}_k, where $k$ is an algebraically closed field of characteristic $p$. The key point is that the presence of Frobenius morphism makes the behavior of rational curves to be very different from that of charateristic 0. We show that if there exists $N_0$ such that for all $e\geq N_0$ there is a degree $e$ very free rational curve on $X$, then $N_0> p^r(p^r-1)$.Comment: 4 page

Sulfur adsorption, structure, and effects on coarsening on Ag(111) and Ag(100)

We reported sulfur adsorption, structure, and effects on coarsening on both Ag(111) and Ag(100) single crystal surfaces. Experiments were performed in ultra high vacuum (UHV) using variable temperature scanning tunneling microscopy (VT-STM) and Auger electron spectroscopy (AES). Sulfur was deposited prior-to, during, and after the deposition of Ag at temperatures usually below 300 K. Comparison with clean Ag(111) and Ag(100) surfaces provides a way to determine how sulfur affects the atomic-scale mass transport on both surfaces. Experimental data show that a well-ordered, self-organized dot-row structure appears after adsorption of S on Ag(111) at 200 K. This dot-row motif, which exhibits fixed spacing between dots within rows, is present over a wide range of coverage. The dots are probably Ag3S3 clusters with adsorbed S in the spaces between dots. Dynamic rearrangements are observed. Trace amounts of adsorbed S below a critical coverage on the order of 10 millimonolayers have little effect on the coarsening and decay of monolayer Ag adatom islands on Ag(111) at 300 K. In contrast, above this critical coverage, coarsening is greatly accelerated. This critical value appears to be determined by whether all S can be accommodated at step edges. Accelerated coarsening derives from the feature that the excess S residing on the terraces produces significant populations of metal-sulfur complexes which are stabilized by strong Ag-S bonding. Furthermore, below room temperature, and at coverages above the threshold, an ordered sulfur structure develops. This structure contains long rows of Ag3S3 trimers as its dominant motif, and its development coincides with inhibition of coarsening. We also show that adsorbed S on Ag/Ag(100) causes the ripening mechanism change from island diffusion/coalescence, i.e., Smoluchowski ripening (SR) to Ostwald ripening (OR) at S coverages from 0.03 ML to 0.21 ML at 300 K. The Ag island decay rate for OR increases with increasing S coverage. Ag islands change from square to round with increasing S coverage, and change orientation at S coverage higher than 0.16 ML. No more coarsening occurs when S coverage increases to 0.27 ML, where ordered structures form

Prym-Tjurin Constructions on Cubic Hypersurfaces

In this paper, we give a Prym-Tjurin construction for the cohomology and the Chow groups for a cubic hypersurface. On the space of lines meeting a given rational curve, there is the incidence correspondence. This correspondence induces an action on both of the primitive cohomology and the primitive Chow groups. We first show that this action satisfies a quadratic equation. Then the Abel-Jacobi homomorphism induces an isomorphism between the primitive cohomology of the cubic hypersurface and the Prym-Tjurin part of the above action. This also holds for Chow groups with rational coefficients

The motive of the Hilbert cube

The Hilbert scheme $X^{[3]}$ of length-$3$ subschemes of a smooth projective variety $X$ is known to be smooth and projective. We investigate whether the property of having a multiplicative Chow-Kuenneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the map $X^3 \dashrightarrow X^{[3]}$. The case of the Hilbert square was taken care of in previous work of ours. The archetypical examples of varieties endowed with a multiplicative Chow-Kuenneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKaehler varieties share the same property. Roughly, if a smooth projective variety $X$ has a multiplicative Chow-Kuenneth decomposition, then the Chow rings of its powers $X^n$ have a filtration, which is the expected Bloch-Beilinson filtration, that is split