What are the consequences in dying as a Hardcore?

Dying as a Hardcore in Diablo 2 looks different, in comparison with dying as a Hardcore, in Diablo 3.

Can someone provide the principal consequences in dying as a Hardcore in Diablo 3?

extra: Can I delete, from my heroes list, my hardcore hero? or must I keep him there for some reason?

Solution 1:

Hardcore is actually very similar to Diablo II, there are a few differences but dying isn't one of them. Death is still permanent.

The main differences are:

• Hardcore characters do not have access to the cash-money auction house. This is believed by most to an effort to limit the economic chaos that will result from folks botting the game to get money in the "softcore" mode.
• You have to reach level 10 with a regular character, before you can start a hardcore character.
• Dead Hardcore characters above level 10 can be "archived" to the "Hall of Fallen Heroes", where you can see a few stats about them. This action also removes them from the slots of your character selection screen.

The video below does a good job of summarizing most of the effects of death as a hardcore character in Diablo III.

Behaviors returning from Diablo II

• Hardcore players can only team with other Hardcore players.
• Your ghost is still able to chat after you die, but you will not ever be able to play the character again.
• Hardcore characters have a red title that is unique from Softcore players

Solution 2:

When your hardcore hero dies, you can no longer play with that hero, and you lose all the items the hero was carrying at the time of death, even if you play with friends (your corpse cannot be looted).

As far as I know, you do not lose anything which is shared - so you do not lose items stored in your chest, or artisan progress, or gold, etc.

A dead hardcore hero does not occupy a hero slot (after you remove them, as per Ebongo's excellent answer).