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--- blog:2023-02-10 [2023/02/10 20:17] – created pzhou
+++ blog:2023-02-10 [2023/06/25 15:53] (current) – external edit 127.0.0.1
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 ====== 2023-02-10 ======
-  * Reading about symmetric product, by [[https://lekili.duckdns.org/research/higheraus.pdf | DJK]] and [[https://arxiv.org/pdf/1003.2962.pdf | Auroux's ICM]]
+  * Reading about symmetric product, by [[https://lekili.duckdns.org/research/higheraus.pdf | Dykerhoff-Jasso-Lekili]] and [[https://arxiv.org/pdf/1003.2962.pdf | Auroux's ICM]].
+  * Idea about generation
+===== Upstairs Skeleton =====
+How to understand it? Downstairs, we have $Sym^2(\C^*)$, which we can shrink to $Sym^2(S^1)$. Upstairs, I just don't know what to put over the diagonal. We know the fiber of the diagonal is $\C^* \times \C_u$, and the superpotential is basically the variable $u$, so if we just ask for the fiberwise skeleton, it is zero.
+Same thing, if we ask for the multiplicative case. The fiberwise skeleton of a smooth fiber is $(S^1)^2$, then become subcritical for the special fiber.
+===== Fiber of W =====
+Consider the simple case. $Hilb^2_{hor}(C^* x C)$, where $W = y_1^2 + y_2^2 + (x_1- x_2)/(y_1-y_2)$. We reduce it. Let $A = x_1 + x_2$, $B = (y_1-y_2)^2$, $u = (x_1- x_2)/(y_1-y_2)$, and $u^2 B - A^2 \neq 0$. Superpotential is $W = B + u$.
+Or $Y = {x^2 - y^2 z \neq 0}$ and $W = y + z$.
+We need to consider what is the divisor $x^2 - y^2 z = 0 $.
+The fiber over $w$ is $C^2 \ \{x^2 - y^2 (w-y) = 0 \}$
+Is that deleted guy smooth? Namely
+$$ xdx - (2w y - 3y^2) dy = 0 $$
+means $x=0, y=0$ or $x=0, y= 2w/3$. So, what happens here? when $w=0$, the two critical points in the deleted divisor merge. What do you want to delete to make the complement the vanishing cycle the same?
+Fiber better be R^2