Study of Immunotherapy Plus ADI-PEG 20 for the Treatment of Advanced Uveal Melanoma

Inclusion Criteria: * Advanced or unresectable melanoma of presumed uveal origin. Non-uveal melanomas with "malignant blue nevus" physiology with GNAQ, GNA11, CYSLTR2, or PLCB4 driver alterations are eligible upon discussion with the Principal Investigator. * Disease must be measurable according to RECIST 1.1. Disease that has undergone local therapy in the past 30 days is not considered measurable unless the investigator has documented progression despite the local therapy. * Disease must be amenable to a biopsy attempt, in the opinion of the investigator. * Asymptomatic untreated brain metastases are allowed. Symptomatic brain metastases that have undergone local therapy with RT or surgery and have not required an increase in steroid dose in prior 2 weeks are allowed. Note: Seizure prophylaxis with untreated brain metastases are allowed. * Patients must have an Easter Cooperative Oncology Group (ECOG) Performance Statue (PS) of 0-1. * Acceptable liver, renal, and hematological function: * Total bilirubin \</= 1.5x upper limit of normal (ULN); patients with Gilbert's Syndrome must have bilirubin \</= 3x ULN * Aspartate aminotransferase (AST/SGOT) and alanine aminotransferase (ALT/SGPT) \</= 3 x ULN (\</= 5x if liver metastases are present) * Estimated glomerular filtration rate (GFR) \>/= 30 mL/min using a cancer-specific GFR Model; the calculator found at: http://tavarelab.cruk.com.ac.uk/JanowitzWilliamsGFR/ * Hemoglobin \>/= 9 g/dL * Neutrophils \>/= 1.5 x 10\^9/L * Platelets \>/= 100 x 10\^9/L * Female patients of childbearing potential and their partners (if male) and male patients with female partners of childbearing potential and their partners must agree to use a highly effective form of contraception for the duration of the study from the list below or agree to refrain from intercourse for the duration of the trial and for at least 30 days after the last administration of ADI-PEG20 and at least 150 days (if female) or 210 days (if male) after the final dose of ipilimumab and/or nivolumab whichever is later. Highly effective forms of contraception include the following: * combined (estrogen and progestogen containing) hormonal contraception associated with inhibition of ovulation (oral, intravaginal or transdermal) * progestogen-only hormonal contraception associated with inhibition of ovulation (oral, injectable, implantable) * intrauterine device * intrauterine hormone-releasing system * bilateral tubal occlusion * vasectomized partner Exclusion Criteria: * Other active malignancy that in the opinion of the treating investigator will interfere with the assessments of efficacy for uveal melanoma in this study * History of seizure disorder not related to malignancy * Pregnancy or lactation * Expected non-adherence to protocol * Known allergy to E. coli drug products (such as GM-CSF) * Known allergy to pegylated compounds * Prior treatment with ADI-PEG20 or another experimental arginine deprivation strategy * Systemic anticancer therapy within 3 weeks or 1 cycle length, whichever is shorter, of first day of planned study therapy * Presence of treatment-related adverse events that have not recovered or stabilized at Grade 1 (excepting vitiligo and alopecia or treated endocrine conditions). AEs that are Grade 2 that are not felt to be a significant safety risk (e.g. rash, asymptomatic thyroiditis) may be allowable upon discussion with the Principal Investigator. * Active autoimmune disease or any condition requiring greater than 10mg prednisone per day equivalent or other immune suppressive medication (e.g. anti-TNF agents) within 14 days of study screening. Inhaled or topical steroids and adrenal replacements does of steroids \>10mg prednisone are allowed in the absence of active autoimmune disease. * History of myocarditis or motor neuropathy of any grade * Gout flares within the past 28 days or sequelae of chronic gout, such as gouty arthritis, are excluded. Note: Patients with asymptomatic hyperuricemia without arthralgias or arthritic symptoms are eligible, as are patients with known gout on chronic uric acid lowering medication who have not experienced a flare within 28 days