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If I have critical specialization with Picks, how much damage do I cause?

Pick critical specialization:

2 additional damage per weapon damage die.

However, Picks also have the Fatal property:

On a critical hit, [...] the weapon adds one additional damage die of the listed size.

So it seems that a puny Pick does 1d6+4 damage on a regular hit, but (1d10+4)*2+1d10+4 on a crit.
Is that right?

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From the CRB, page 279:

Counting Damage Dice

Effects based on a weapon's number of damage dice include only the weapon's damage die plus any extra dice from a striking rune. The don't count extra dice from abilities, critical specialisation effects, property runes, weapon traits, or the like.

So the extra die from the weapon trait Fatal does not increase the number of dice for the purpose of calculating the critical specialisation effect, i.e. you get +2 damage for a non-magical pick, +2 per die from a striking rune.

Your example Pick would deal (1d10+4)*2 +1d10 +2 on a critical hit.

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For a nonmagical pick, you deal (d6+static modifiers) on a regular hit, (d10+static modifiers)x2+2+d10 on a critical hit

Where "static modifiers" includes things like your strength modifier and your weapon specialization damage bonus. This assumes you have the critical specialization for Picks.

CRB page 451 says the following about doubling on critical hits:

"Benefits you gain specifically from a critical hit, like the flaming weapon rune’s persistent fire damage or the extra damage die from the fatal weapon trait, aren’t doubled."

The pick critical specialization grants a benefit on a critical hit, so it isn't multiplied by 2 for the critical. The fatal weapon trait increases all damage dice to d10s, but the bonus damage die is a benefit gained from a critical hit so it also isn't doubled.

As a further example: If your pick is a striking weapon, its base weapon dice are 2d6, so you'd deal (2d6+static modifiers) on a regular hit, and (2d10+static modifiers)x2+4+d10 on a critical hit.

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    \$\begingroup\$ The question wasn't about doubling, but the 2 extra damage from Pick critical specialization \$\endgroup\$ – András Nov 8 '19 at 22:20

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