A consequence of Germany promising Austria a "blank check" was that:.

The consequence was Austria's move to end all anti-Austrian movements in Serbia.

Explanation:

The "blank check" was a promise made by Germany to Austria. In that promise, the German government pledged to help Austria fight the anti-Austrian movements that were settling in Serbia. For Germany, it was important that no one interfered in the conflict between Austria and Serbia and therefore promised to protect Austria, even if Russia tried to interfere.

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