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We use high-resolution N-body simulations to study the galaxy-clustercross-sections and the abundance of giant arcs in the $\Lambda$CDM model.Clusters are selected from the simulations using the friends-of-friends method,and their cross-sections for forming giant arcs are analyzed. The backgroundsources are assumed to follow a uniform ellipticity distribution from 0 to 0.5and to have an area identical to a circular source with diameter $1\arcsec$. Wefind that the optical depth scales as the source redshift approximately as$\tau_{1''} = 2.25 \times 10^{-6}/[1+(\zs/3.14)^{-3.42}]$ ($0.6<\zs<7$). Theamplitude is about 50% higher for an effective source diameter of $0.5\arcsec$.The optimal lens redshift for giant arcs with the length-to-width ratio ($L/W$)larger than 10 increases from 0.3 for $\zs=1$, to 0.5 for $\zs=2$, and to0.7-0.8 for $\zs>3$. The optical depth is sensitive to the source redshift, inqualitative agreement with Wambsganss et al. (2004). However, our overalloptical depth appears to be only $\sim$ 10% to 70% of those from previousstudies. The differences can be mostly explained by different power spectrumnormalizations ($\sigma_8$) used and different ways of determining the $L/W$ratio. Finite source size and ellipticity have modest effects on the opticaldepth. We also found that the number of highly magnified (with magnification$|\mu|>10$) and ``undistorted'' images (with $L/W<3$) is comparable to thenumber of giant arcs with $|\mu|>10$ and $L/W>10$. We conclude that ourpredicted rate of giant arcs may be lower than the observed rate, although theprecise `discrepancy' is still unclear due to uncertainties both in theory andobservations.Comment: Revised version after the referee's reports (32 pages,13figures). The  paper has been significantly revised with many additions. The new version  includes more detailed comparisons with previous studies, including the  effects of source size and ellipticity. New discussions about the redshift  distribution of lensing clusters and the width of giant arcs have been adde

Is the Number of Giant Arcs in ΛCDM Consistent with Observations?