DNA sequencing offers a powerful tool in oncology based on the precise definition of structural rearrangements and copy number in tumor genomes. Here, we describe the development of methods to compute copy number and detect structural variants to locally reconstruct highly rearranged regions of the tumor genome with high precision from standard, short-read, paired-end sequencing datasets. We find that circular assemblies are the most parsimonious explanation for a set of highly amplified tumor regions in a subset of glioblastoma multiforme samples sequenced by The Cancer Genome Atlas (TCGA) consortium, revealing evidence for double minute chromosomes in these tumors. Further, we find that some samples harbor multiple circular amplicons and, in some cases, further rearrangements occurred after the initial amplicon-generating event. Fluorescence in situ hybridization analysis offered an initial confirmation of the presence of double minute chromosomes. Gene content in these assemblies helps identify likely driver oncogenes for these amplicons. RNA-seq data available for one double minute chromosome offered additional support for our local tumor genome assemblies, and identified the birth of a novel exon made possible through rearranged sequences present in the double minute chromosomes. Our method was also useful for analysis of a larger set of glioblastoma multiforme tumors for which exome sequencing data are available, finding evidence for oncogenic double minute chromosomes in more than 20% of clinical specimens examined, a frequency consistent with previous estimates.