Assume we have a normal,connected quasi projective scheme $Y:=X\backslash D$ where $X$ is a quasi projective scheme over field $k$, not necessarily char zero and also $D$ is a simple divisor, not necessarily connected. Then assume $W$ is a connected etale covering of $Y$ and $C$ is a smooth connected curve inside $X$ which intersects $D$ transversally. Then can one say $W \times_X C$ is connected?
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$\begingroup$ Do you mean $X \setminus D$? Also, what makes a divisor simple? Are you assuming that $W$ is connected (otherwise there is no hope)? Even so, assuming I'm reading this right, it seems very unlikely to be true. Do you have any reason to believe it's true? Have you tried any examples? $\endgroup$– Karl SchwedeCommented Dec 21, 2014 at 4:32
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$\begingroup$ corrected, simple means $D=\cup D_i$ such that $D_i$ are effective cartier divisor and they don't intersect. $\endgroup$– Sina BaghalCommented Dec 21, 2014 at 4:39
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Ok, I'm pretty sure this is false but I could be misreading so let's try a really easy example. Assuming I'm reading this correctly, it might be more appropriate for math stackexchange.
Say $X$ is the blowup of $\mathbb{P}^2_k$ at $[0,1,0]$ where $k = \overline{k}$ is not of characteristic 2. Say that $D$ is the simple divisor corresponding to the strict transform of the line at infinity $[s,t,0]$ and the strict transform $y$-axis on the remaining chart $[0,s,t]$. These two divisors intersected on $\mathbb{P}^2$ but not on the blowup. Note $Y= \text{Spec } k[x,y, x^{-1}]$ is $\mathbb{A}^2 \setminus (y-\text{axis})$.
Consider $W \to Y$ corresponding to the inclusion: $$k[x^{1/2}, y, x^{-1/2}] \subseteq k[x,y, x^{-1}].$$ This is obviously etale. Finally let $C$ be the curve corresponding to the closure of $V(x-1)$ (it goes and intersects $D$ transversaly at infinity). Then obviously $W \times_X C = V( (x^{1/2} - 1)(x^{1/2}+1) )$ which is clearly not connected.
answered Dec 21, 2014 at 4:43
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$\begingroup$ There are much simpler counterexamples. Just take $X$ to be $A\times B$, a product of projective smooth curves. Let $D$ be the empty set; if the OP insists that $D$ is nonempty, let $D$ be $A \times \{b_0\}$. Let $\widetilde{B}\to B$ be a finite etale cover, and let $W$ be $A\times \widetilde{B}$ with its associated morphism to $X$ (remove the inverse image of $D$ if $D$ is nonempty). Let $C$ be $A\times\{b\}$ for general $b$. $\endgroup$ Commented Dec 21, 2014 at 11:41
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1$\begingroup$ Hi Jason, the first example I wrote was with $D$ empty (if you look at the revision history), but then I thought that the op would probably want something with $D$ non-empty and with non-empty transverse intersection with $C$. $\endgroup$ Commented Dec 21, 2014 at 15:58