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in what way is crossing over important for sexual reproduction?
It's not important in the sense that sexual reproduction can still happen even if crossing over doesn't.
Where its importance comes into play is by allowing an even greater variation in the genetic diversity of the offspring. The easiest way to see this is with an example. I'll use one with 5 traits on one chromosome. Here are the traits:
Y=yellow seed, y=green seed
R=round, r=wrinkled
T=tall, t=short
S=smooth, s=wrinkled
P=purple flower, p=white flower
Suppose now that you had a parent YyRrTtSsPp. You should know that since the fgenes are all on the same chromosome, the parent can only produce gametes that are YRTSP or yrtsp. At least that's what happens without crossing over. But what can happen with crossing over, is that when the sister chromatids line up during Meiosis I, the inner two chromatids can have parts exchange places. So you could get a chromosome with YRTsp and the other yrtSP as well as the outer two which would still be YRTSP or yrtsp. To see this pictorially:
Y Y y y
R R r r
ToT tot
S S s s
P P p p <<-- without crossing over, the "o" is the centromere holding the sister chromatids together
Y Y y y
R R r r
ToT tot
S s Ss
P p p P <<-- with crossing over - note the center DNA strands
So with crossing over, you get more possible gene combinations.
