Intersection positivity for curves and surfaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:09:44Z http://mathoverflow.net/feeds/question/79273 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79273/intersection-positivity-for-curves-and-surfaces Intersection positivity for curves and surfaces Parsa 2011-10-27T14:41:20Z 2011-10-28T07:43:52Z <p>Let $X$ be a smooth complete variety over an algebraically closed field of dimension $\geq3$. Given a divisor $D_1$ on $X$ with $D_1 \cdot C>0$ for every curve $C \subset X$, and a divisor $D_2$ on $X$ satisfying $D_2^2 \cdot S>0$ for every surface $S \subset X$, does there exist a divisor $D$ on $X$ satisfying $D \cdot C>0$ <em>and</em> $D^2 \cdot S>0$ for every curve and surface respectively? I am willing to make any assumption on $X$, except that $X$ be projective.</p> <p>As I understand it, since $D_1$ is nef, we have that $D_1^2 \cdot S\geq 0$, so even if for $m>>0$ we manage to have $(mD_1+D_2) \cdot C>0$ for every curve, the other requirement becomes $(mD_1+D_2)^2 \cdot S = m^2(D_1^2 \cdot S) + 2m(D_1\cdot D_2 \cdot S) + D_2^2 \cdot S >0$. The first term is non-negative, the last term is positive, but what needs to happen to ensure the middle term is non-negative also?</p> http://mathoverflow.net/questions/79273/intersection-positivity-for-curves-and-surfaces/79359#79359 Answer by Sándor Kovács for Intersection positivity for curves and surfaces Sándor Kovács 2011-10-28T07:43:52Z 2011-10-28T07:43:52Z <p>I am not sure whether your original question is true, but it seems to me that your proposed solution does not work. Here is why:</p> <p>The main problem is that you know very little about $D_2$. Given what we know, it does not even have to be effective!</p> <p><strong>Example</strong> Let $X$ be a smooth complete variety with divisors $A,B$ such</p> <ul> <li><p>$A\cdot B=0$</p></li> <li><p>$(A^2+B^2)\cdot S>0$ for any surface $S$, and</p></li> <li><p>$B$ and $B^2$ are effective classes.</p></li> </ul> <p>Then $D_2=A-B$ has the property that $D_2^2\cdot S>0$ for any $S$, but $D_1\cdot D_2\cdot B\cdot H &lt;0$ for any $D_1$ divisor satisfying the criterion given in the question and an appropriate $H$ such that $B\cdot S$ is an effective surface and $B^2\cdot H$ is an effective curve.</p> <p>To make this complete I should give an example that such an $X$ with $A,B$ exists. I think I can construct an example (a product of two surfaces blown up along a surface) satisfying these properties, but I don't know a simple one. In any case I think it is reasonable to expect that such an example exists which shows that $D_1\cdot D_2\cdot S$ is not necessarily non-negative even on a projective variety.</p>