Half your original basis "steps up" to the fair market value (FMV) on the date of death. Using an example, you jointly owned 100 shares of stock bought in 1999 for $1000. When your spouse passed in 2020, the stock was worth $3000. Your new basis is $2000: $500 (your half of the original $1000 cost) + $1500 (spouse's half of the $3000 FMV).
If you live in a community property state, the entire basis steps up to the FMV on the date of death.
