$unwind empty array

With MongoDB 3.2 version and newer, the $unwind operator now has some options where in particular the preserveNullAndEmptyArrays option will solve this.

If this option is set to true and if the path is null, missing, or an empty array, $unwind outputs the document. If false, $unwind does not output a document if the path is null, missing, or an empty array. In your case, set it to true:

db.collection.aggregate([
    { "$unwind": {
        "path": "$solved",
        "preserveNullAndEmptyArrays": true
    } },
    { "$group": {
        "_id": "$_id",
        "login": { "$first": "$login" },
        "solved": { "$sum": "$solved.points" }
    } }
])

Here is the solution - it assumes that the field "solved" is either absent, is equal to null or has an array of problems and scores solved. The case it does not handle is "solved" being an empty array - although that would be a simple additional adjustment you could add.

project = {$project : {
        "s" : {
            "$ifNull" : [
                "$solved",
                [
                    {
                        "points" : 0
                    }
                ]
            ]
        },
        "login" : 1
    }
};
unwind={$unwind:"$s"};
group= { "$group" : {
        "_id" : "$_id",
        "login" : {
            "$first" : "$login"
        },
        "score" : {
            "$sum" : "$s.points"
        }
    }
}

db.students.aggregate( [ project, unwind, group ] );