Numerically rigid nef divisor

2010-11-16T11:27:04Z

Is it possible to find an example of an $\mathbb{R}$-Cartier divisor $D$ on an irreducible variety $X$ that is non-trivial, nef, effective and numerically rigid?

By "numerically rigid" I mean that if $E$ is another $\mathbb{R}$-Cartier effective divisor such that $E$ is numerically equivalent to $D$ then $D=E$.

For curves this clearly cannot be the case, since an effective non-trivial divisor is always ample.

Answer by Francesco Polizzi for Numerically rigid nef divisor

2010-11-16T11:53:32Z

Take a minimal surface $S$ of general type with $p_g=1$, $q=0$ and zero torsion.

Then $S$ contains a unique effective canonical curve $K$, which is nef and numerically rigid.

In fact, since $q=0$ and there is no torsion, we have $\textrm{Pic}^0(S)=0$, the Neron - Severi group $\textrm{NS}(S)$ coincides with the Picard group $\textrm{Pic}(S)$ and any two numerically equivalent divisors on $S$ are linearly equivalent.

Examples of these surfaces, with $K^2=2$, are given in the paper of Debarre and Catanese

"Surfaces with $K^2=2$, $p_g=1$, $q=0$",

J. reine angew. Math. 395 (1989), 1-55.

Numerically rigid nef divisor - MathOverflow

Numerically rigid nef divisor

Answer by Francesco Polizzi for Numerically rigid nef divisor